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Reoptimization techniques for solving hard problems

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Jhoirene Clemente

The document discusses reoptimization techniques for solving hard optimization problems, highlighting the classification of such problems and their complexity, particularly focusing on NP-hard issues like the Traveling Salesman Problem (TSP). It details definitions, problem structures, and various approaches for tackling these challenges, including exact algorithms, approximation algorithms, and heuristics. Several optimization problems are explored along with methodologies such as dynamic programming and greedy algorithms.

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Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines Diliman Candidacy Exam July 30, 2014
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 2 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 3 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma Conditions
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 4 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 5 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 6 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 7 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 8 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hamiltonian Cycle Definition (Hamiltonian Cycle) A Hamiltonian cycle is a cycle in a graph passing through all the vertices once.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 9 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 10 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 11 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 12 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 13 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 14 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 15 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 16 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveling Salesman Problem (TSP) Definition INPUT: Edge weighted Graph G = (V,E) OUTPUT: Minimum cost Hamiltonian Cycle.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 17 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveling Salesman Problem (TSP) Definition INPUT: Edge weighted Graph G = (V,E) OUTPUT: Minimum cost Hamiltonian Cycle.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 18 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Optimization Problems [Papadimitriou and Steiglitz, 1998] Definition (Instance of an Optimization Problem) An instance of an optimization problem is a pair (F, c), where F is any set, the domain of feasible points; c is the cost function, a mapping c : F ! R The problem is to find an f 2 F for which c(f ) c(y) 8 y 2 F Such a point f is called a globally optimal solution to the given instance, or, when no confusion can arise, simply an optimal solution.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 18 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Optimization Problems [Papadimitriou and Steiglitz, 1998] Definition (Instance of an Optimization Problem) An instance of an optimization problem is a pair (F, c), where F is any set, the domain of feasible points; c is the cost function, a mapping c : F ! R The problem is to find an f 2 F for which c(f ) c(y) 8 y 2 F Such a point f is called a globally optimal solution to the given instance, or, when no confusion can arise, simply an optimal solution. Definition (Optimization Problem) An optimization problem is a set of instances of an optimization problem.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 19 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Optimization Problems [Papadimitriou and Steiglitz, 1998] Two categories 1. with continuous variables 2. with discrete variables, which we call combinatorial
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 20 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Combinatorial Optimization Problem [Papadimitriou and Steiglitz, 1998] Definition (Combinatorial Optimization Problem ) An optimization problem consists of = (D,R, cost, goal) 1. D: A set of all valid input instances. Input instance I 2 D 2. R: A set of all feasible solution. Each I 2 D has a set of feasible solutions, R(I ). 3. cost: Objective function that assigns a non-negative rational number to each pair (I , SOL) cost(I , SOL), where I 2 D and SOL 2 R(I ). 4. goal 2 {min, max}.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 21 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 22 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
Reoptimization techniques for solving hard problems
Reoptimization techniques for solving hard problems
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 25 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Growth of N!
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable!
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable! TSP is NP-hard!
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable! TSP is NP-hard! Unless P=NP, we cannot obtain an efficient algorithm solving the problem.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable! TSP is NP-hard! Unless P=NP, we cannot obtain an efficient algorithm solving the problem.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 27 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity NP-hard Problems Problem A is NP-hard if all problems in NP are polynomial-time reducible to it. NP-hard problems are at least as hard as the hardest problems in NP.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 28 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 28 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 28 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes I P is the class of decision problems solvable in polynomial time by a Turing machine (TM). I NP is the class of decision problems verifiable in polynomial time by a Turing Machine. I A decision problem is NP-complete if (1) it is in NP, and (2) any problem in NP can be reduced to it (under some notion of reduction). The class of NP-complete problems is sometimes called NPC.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 29 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Examples of NP-hard Problems [Garey and Johnson, 1979] 1. Satisfiability problem 2. Clique Problem 3. Independent set problem 4. Vertex Cover Problem 5. Traveling salesman problem 6. Median String Problem 7. and many more
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 30 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems Design technique have a well specified structure that even provides a framework for possible implementations Concepts formulate ideas and rough frameworks about how to attack hard algorithmic problems .
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 30 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems Design technique have a well specified structure that even provides a framework for possible implementations I Dynamic Programming I Branch and Bound I Greedy Schema I Genetic Algorithm I Simulated Annealing, Tabu Search I Local search I . . . Concepts formulate ideas and rough frameworks about how to attack hard algorithmic problems . I Approximation Algorithms I Parameterized Algorithms I Randomized Algorithms I . . .
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 31 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems I Exact algorithm always obtain the optimal solution I Approximation algorithm settle for good enough solutions. Goodness of solution is guaranteed and measured using approximation ratio. I Heuristic algorithms produce solutions, which are not guaranteed to be close to the optimum. The performance of heuristics is often evaluated empirically.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 32 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems I Exact algorithm always obtain the optimal solution I Approximation algorithm settle for good enough solutions. Goodness of solution is guaranteed and measured using approximation ratio. I Heuristic algorithms produce solutions, which are not guaranteed to be close to the optimum. The performance of heuristics is often evaluated empirically.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 33 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Triangle inequality Edge weighted graph with cost function c satisfies triangle inequality if for all u, v, and w 2 V c(u, w) c(u, v) + c(v, w) .
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 34 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Triangle inequality c(u, w) c(u, v) + c(v, w)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 35 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Solving Traveller’s Dilemma INPUT: Edge weighted graph 1. Compute a Minimum Spanning Tree for G 2. Select a vertex r 2 V(G) to be the root vertex 3. Let L be the list of vertices visited in a preorder walk OUTPUT: Hamiltonian Cycle that visits A preorder tree walk recursively visits every vertex in the tree, listing a vertex when it is first encountered , before any of its children are visited.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 36 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 37 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 38 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 39 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 40 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 41 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 42 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 43 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 44 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 45 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 46 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 47 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 48 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 49 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 50 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 51 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 52 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 53 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 54 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hamiltonian Cycle from the Preorder Walk
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 55 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hamiltonian Cycle from the Preorder Walk 4 + 4 + 10 + 15 + 10 + 16 = 59
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 56 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Total tour cost vs. the optimal cost Let H denote an optimal Hamiltonian tour in G. Let T be the minimum spanning tree of G. c(T) c(H) Let W be the full length cost of the preorder walk. c(W) = 2c(T) Let H be the output Hamiltonian Cycle from the algorithm c(W) c(H) Then c(H) 2c(H)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 56 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Total tour cost vs. the optimal cost Let H denote an optimal Hamiltonian tour in G. Let T be the minimum spanning tree of G. c(T) c(H) Let W be the full length cost of the preorder walk. c(W) = 2c(T) Let H be the output Hamiltonian Cycle from the algorithm c(W) c(H) Then c(H) 2c(H) Approximation ratio () is 2
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 57 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Algorithms Definition (Approximation Algorithm [Williamson and Shmoy, 2010] ) An -approximation algorithm for an optimization problem is a polynomial-time algorithm that for all instances of the problem produces a solution whose value is within a factor of of the value of an optimal solution. Given a problem instance I with an optimal solution Opt(I ), i.e. the cost function cost(Opt(I )) is minimum/maximum.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 57 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Algorithms Definition (Approximation Algorithm [Williamson and Shmoy, 2010] ) An -approximation algorithm for an optimization problem is a polynomial-time algorithm that for all instances of the problem produces a solution whose value is within a factor of of the value of an optimal solution. Given a problem instance I with an optimal solution Opt(I ), i.e. the cost function cost(Opt(I )) is minimum/maximum. I An algorithm for a minimization problem is called -approximative algorithm for some 1, if the algorithm obtains a maximum cost of · cost(Opt(I )), for any input instance I .
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 57 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Algorithms Definition (Approximation Algorithm [Williamson and Shmoy, 2010] ) An -approximation algorithm for an optimization problem is a polynomial-time algorithm that for all instances of the problem produces a solution whose value is within a factor of of the value of an optimal solution. Given a problem instance I with an optimal solution Opt(I ), i.e. the cost function cost(Opt(I )) is minimum/maximum. I An algorithm for a minimization problem is called -approximative algorithm for some 1, if the algorithm obtains a maximum cost of · cost(Opt(I )), for any input instance I . I An algorithm for a maximization problem is called -approximative algorithm, for some 1, if the algorithm obtains a minimum cost of · cost(Opt(I )), for any input instance I .
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 58 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio I Minimization, 1 cost(Opt(I )) cost(SOL) cost(Opt(I )) I Maximization, 1 cost(Opt(I )) cost(SOL) cost(Opt(I ))
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 59 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio I Additive Approximation Algorithms: SOL OPT + c I Constant Approximation Algorithms: SOL c · OPT I Logarithmic Approximation Algorithms: SOL = O(log n) · OPT I Polynomial Approximation Algorithms: SOL = O(nc) · OPT, where c 1
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 60 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximable Problems [Vazirani, 2001] Definition (APX) I An abbreviation for “Approximable, is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 60 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximable Problems [Vazirani, 2001] Definition (APX) I An abbreviation for “Approximable, is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant. I Problems in this class have efficient algorithms that can find an answer within some fixed percentage of the optimal answer.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 61 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Polynomial-time Approximation Schemes Definition (PTAS [Aaronson et al., 2008]) The subclass of NPO problems that admit an approximation scheme in the following sense. For any 0, there is a polynomial-time algorithm that is guaranteed to find a solution whose cost is within a 1 + factor of the optimum cost. Contains FPTAS, and is contained in APX.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 61 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Polynomial-time Approximation Schemes Definition (PTAS [Aaronson et al., 2008]) The subclass of NPO problems that admit an approximation scheme in the following sense. For any 0, there is a polynomial-time algorithm that is guaranteed to find a solution whose cost is within a 1 + factor of the optimum cost. Contains FPTAS, and is contained in APX. Definition (FPTAS [Aaronson et al., 2008] ) The subclass of NPO problems that admit an approximation scheme in the following sense. For any 0, there is an algorithm that is guaranteed to find a solution whose cost is within a 1 + factor of the optimum cost. Furthermore, the running time of the algorithm is polynomial in n (the size of the problem) and in 1/.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 62 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio of well known Hard Problems 1. FPTAS: Bin Packing Problem 2. PTAS: Makespan Scheduling Problem 3. APX: 3.1 Min Steiner Tree Problem (1.55 Approximable) [Robins,2005] 3.2 Min Metric TSP (3/2 Approximable) [Christofides,1977] 3.3 Max SAT (0.77 Approximable) [Asano,1997] 3.4 Vertex Cover (2 Approximable) [Vazirani,2001] 4. MAX SNP 4.1 Independent Set Problem 4.2 Clique Problem 4.3 Travelling Salesman Problem
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 63 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes [Ausiello et al., 2011] FPTAS ( PTAS ( APX ( NPO
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 64 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Inapproximable Problems Many problems have polynomial-time approximation schemes. However, there exists a class of problems that is not so easy [Williamson and Shmoy, 2010]. This class is called MAX SNP. Theorem For any MAXSNP-hard problem, there does not exist a polynomial-time approximation scheme, unless P = NP [Williamson and Shmoy, 2010].
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 64 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Inapproximable Problems Many problems have polynomial-time approximation schemes. However, there exists a class of problems that is not so easy [Williamson and Shmoy, 2010]. This class is called MAX SNP. Theorem For any MAXSNP-hard problem, there does not exist a polynomial-time approximation scheme, unless P = NP [Williamson and Shmoy, 2010].
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 64 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Inapproximable Problems Many problems have polynomial-time approximation schemes. However, there exists a class of problems that is not so easy [Williamson and Shmoy, 2010]. This class is called MAX SNP. Theorem For any MAXSNP-hard problem, there does not exist a polynomial-time approximation scheme, unless P = NP [Williamson and Shmoy, 2010]. Theorem If P6= NP, then for any constant 1, there is no polynomial-time approximation algorithm with approximation ratio for the general travelling salesman problem.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 65 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: with additional informa-tion
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 66 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: with additional informa-tion
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 67 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: with additional informa-tion
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 68 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 69 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 70 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 71 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009] 10 + 12 + 6 + 4 + 5 + 11 = 48
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 72 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Comparison
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 72 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Comparison 10+12+6+4+5+11 = 48
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 72 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Comparison 10+12+6+4+5+11 = 48 4+4+10+15+10+16 = 59
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 73 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 73 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1 Theorem There is a 3/4 Approximation algorithm for solving Metric TSP.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 74 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity 3/2- Approximation Algorithm [Ausiello et al., 2009] Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1 Algorithm: Output the best solution between (H1,H2), where H1 =Nearest Insert(In,H n , In+1) H2 = 32 -Approximation Algorithm(In+1)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 74 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity 3/2- Approximation Algorithm [Ausiello et al., 2009] Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1 Algorithm: Output the best solution between (H1,H2), where H1 =Nearest Insert(In,H n , In+1) H2 = 32 -Approximation Algorithm(In+1) Proof. n+1) + 2d(v, n + 1) Triangle Inequality ) c(H1) c(H c(H1) c(H n+1) + 2max(d(i, n + 1), d(j, n + 1))) c(H2) 3/2c(H n+1) − max(d(i, n + 1), d(j, n + 1))) min(c(H1), c(H2)) (1/3)c(H1) + (2/3)c(H2) 4/3c(H n+1)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 75 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization [Ausiello et al., 2009] The concept of reoptimization can in many cases help to improve the approximability of hard optimization problems by lowering constant approximation ratios or even constructing a PTAS for problems which are APX-hard in their classical non reoptimization variant.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 76 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Don’t start from scratch when confronted with a problem, but try to make good use of prior knowledge about similar problem instances whenever they are available.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 76 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Don’t start from scratch when confronted with a problem, but try to make good use of prior knowledge about similar problem instances whenever they are available. Definition (Reoptimization [Hans-joachim Böckenhauer, 2008] ) Given a problem instance and an optimal solution for it, we are to efficiently obtain an optimal solution for a locally modified instance of the problem.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 76 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Don’t start from scratch when confronted with a problem, but try to make good use of prior knowledge about similar problem instances whenever they are available. Definition (Reoptimization [Hans-joachim Böckenhauer, 2008] ) Given a problem instance and an optimal solution for it, we are to efficiently obtain an optimal solution for a locally modified instance of the problem. Definition (Postoptimality Analysis [Hoesel and Wagelmans, 1999]) Postoptimality analysis deals with the related question of how much an instance may be altered without changing the set of optimal solutions.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 77 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 78 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 79 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 80 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 81 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Definition (Reoptimization [Zych, 2012]) Let = (D,R, cost, goal) be an optimization problem and M D × D be a binary relation (the modification). The corresponding reoptimization problem RM() = (DRM(),RRM(), costRM(), goalRM()) consists of 1. a set of feasible instances defined as DRM() = {(I , I 0, SOL) : (I , I 0) 2M and SOL 2 R(I )}; we refer to I as the original instance and to I 0 as the modified instance 2. a feasibility relation defined as RRM()((I , I 0, SOL)) = R(I 0)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 82 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hardness of Reoptimization I NP hardness of a problem implies that the solution to a locally modified problem cannot be found in polynomial-time, given the solution of the original problem. I Reoptimization of a Hard Problem is still NP-hard! [Hans-joachim Böckenhauer, 2008]. Proof. Let be NP-hard. Let RM() be the reoptimization variant of To prove that RM() is NP-hard, we need to show that we can reduce to RM().
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 82 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hardness of Reoptimization I NP hardness of a problem implies that the solution to a locally modified problem cannot be found in polynomial-time, given the solution of the original problem. I Reoptimization of a Hard Problem is still NP-hard! [Hans-joachim Böckenhauer, 2008]. Proof. Let be NP-hard. Let RM() be the reoptimization variant of To prove that RM() is NP-hard, we need to show that we can reduce to RM(). I But it is an advantage to know a solution of a problem instance, as compared to not knowing it.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 83 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization 1. Reoptimization can help in providing efficient algorithms to problems involved in dynamic systems 2. Reoptimization can help in providing a better solution or an efficient algorithm for computationally hard problems
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 83 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization 1. Reoptimization can help in providing efficient algorithms to problems involved in dynamic systems I Shortest Path Problem [Pallottino and Scutella, 2003] [Nardelli et al., 2003] I Dynamic Minimum Spanning Tree [Thorup, 2000] with edge weights [Ribeiro and Toso, 2007] [Cattaneo et al., 2010] I Vehicle Routing Problem [Secomandi and Margot, 2009] I Facility Location Problem [Shachnai et al., 2012] 2. Reoptimization can help in providing a better solution or an efficient algorithm for computationally hard problems I Traveling salesman problem [Královic and Mömke, 2007] [Hans-joachim Böckenhauer, 2008] [Ausiello et al., 2011] I Steiner tree problem [Hromkovic, 2009] [Zych and Bilò, 2011][Bilo and Zych, 2012] [Böckenhauer et al., 2012] I Shortest common superstring [Bilo,2011] [Popov, 2013] I Hereditary problems on Graphs [Boria et al., 2012a] I Scheduling Problem [Boria et al., 2012b] I Pattern Matching [Yue and Tang, 2008] [?]
Reoptimization techniques for solving hard problems
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 85 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems Definition (Self-reducibility [Zych, 2012]) We will say that a problem is self-reducible if there is a polynomial-time algorithm, , satisfying the following conditions. 1. Given an instance I and an atom of a solution to I , outputs an instance I. We require that the size of I is smaller than the size of I , i.e. |I| |I |. Let R(I |) represent the set of feasible solutions to I containing the atom . We require that every solution SOL of I, i.e., SOL 2 R(I), has a corresponding SOL [ {} 2 R(I |) and that this correspondence is one-to-one. 2. For any set H 2 R(I) it holds that the cost(I ,H [ {}) = cost(I , ) + cost(I,H).
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 86 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] An atom, as used in this context, constitutes a solution for a specific problem instance. For example, in a maximum independent set problem, atoms are the vertices of the graph which constitutes the solution.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 87 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Example Weighted Maximum Independent Set is self-reducible G = (V,E) SOL 2 R(G) () (SOL V(G)) and SOL is independent in G Atoms(G) = V(G) Let be the self-reduction function Gv = (G, v)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 87 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Example Weighted Maximum Independent Set is self-reducible G = (V,E) SOL 2 R(G) () (SOL V(G)) and SOL is independent in G Atoms(G) = V(G) Let be the self-reduction function Gv = (G, v)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 87 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Example Weighted Maximum Independent Set is self-reducible G = (V,E) SOL 2 R(G) () (SOL V(G)) and SOL is independent in G Atoms(G) = V(G) Let be the self-reduction function Gv = (G, v) (1) SOL = SOL [ {v}. (2) cost(G, SOL) = cost(Gv, SOLv) + c(v)
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 88 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Definition (Self-reducible problems) Set of combinatorial problems which admits the self-reducibility property (e.g., Max SAT, Min Steiner Tree) Classified different types of modifications 1. Progressing Subtractive 2. Progressing Additive 3. Regressing Subtractive 4. Regressing Additive Table: Summary of approximation ratio from [Zych, 2012]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 89 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 90 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 91 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 92 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012] Table: Summary of approximation ratio from [Zych, 2012]
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 93 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References Aaronson, S., Kuperberg, G., and Granade, C. (2008). The complexity zoo.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 94 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Bilo, D. and Zych, A. (2012). New Advances in Reoptimizing the Minimum Steiner Tree Problem. Lecture Notes in Computer Science, Mathematical Foundations of Computer Science, 7464:184–197. Böckenhauer, H.-J., Freiermuth, K., Hromkovic, J., Mömke, T., Sprock, A., and Steffen, B. (2012). Steiner tree reoptimization in graphs with sharpened triangle inequality. Journal of Discrete Algorithms, 11:73–86. Boria, N., Monnot, J., and Paschos, V. (2012a). Reoptimization of maximum weight induced hereditary subgraph problems. Theoretical Computer Science, pages 1–12.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 95 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Boria, N., Monnot, J., Paschos, V. T., Bilò, D., Widmayer, P., and Zych, A. (2012b). Reoptimization of the Maximum Weighted Pk-Free Subgraph Problem under Vertex Insertion. 5426:76–87. Cattaneo, G., Faruolo, P., Petrillo, U. F., and Italiano, G. (2010). Maintaining dynamic minimum spanning trees: An experimental study. Discrete Applied Mathematics, 158(5):404–425. Garey, M. R. and Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 96 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Hans-joachim Böckenhauer, J. H. T. M. P. W. (2008). On the hardness of reoptimization. In Proc. of the 34th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2008), LNCS, 4910. Hoesel, S. V. and Wagelmans, A. (1999). On the complexity of postoptimality analysis of 01 programs. Discrete Applied Mathematics, (98). Hromkovic, J. (2009). Algorithmic adventures: from knowledge to magic. Královic, R. and Mömke, T. (2007). Approximation Hardness of the Traveling Salesman Reoptimization Problem. MEMICS 2007, 293.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 97 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Nardelli, E., Proietti, G., and Widmayer, P. (2003). Swapping a Failing Edge of a Single Source Shortest Paths Tree Is Good and Fast. Algorithmica, pages 56–74. Pallottino, S. and Scutella, M. (2003). A new algorithm for reoptimizing shortest paths when the arc costs change. Operations Research Letters. Papadimitriou, C. and Steiglitz, K. (1998). Combinatorial optimization: algorithms and complexity. Popov, V. (2013). On Reoptimization of the Shortest Common Superstring Problem. Applied Mathematical Sciences, 7(24):1195–1197.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 98 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Ribeiro, C. and Toso, R. (2007). Experimental analysis of algorithms for updating minimum spanning trees on graphs subject to changes on edge weights. Experimental Algorithms. Secomandi, N. and Margot, F. (2009). Reoptimization Approaches for the Vehicle-Routing Problem with Stochastic Demands. Shachnai, H., Tamir, G., and Tamir, T. (2012). A Theory and Algorithms for Combinatorial Reoptimization. Lecture Notes in Computer Science, 7256(1574):618–630. Thorup, M. (2000). Dynamic Graph Algorithms with Applications. Proceedings of the 7th Scandinavian Workshop on Algorithm Theory, pages 1–9.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 99 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Vazirani, V. V. (2001). Approximation Algorithms, volume 94. Williamson, D. and Shmoy, D. (2010). The Design of Approximation Algorithms. Yue, F. and Tang, J. (2008). A new approach for tree alignment based on local re-optimization. In International Conference on BioMedical Engineering and Informatics, BMEI 2008. Zych, A. (2012). Reoptimization of NP-hard Problems. PhD thesis, ETH Zurich.
103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization 100 Other Results on Reoptimization Heredity Problems References Algorithms and Complexity References (cont.) Zych, A. and Bilò, D. (2011). New reoptimization techniques applied to Steiner tree problem. Electronic Notes in Discrete Mathematics.

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Reoptimization techniques for solving hard problems

  • 1. Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines Diliman Candidacy Exam July 30, 2014
  • 2. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 2 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma
  • 3. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 3 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma Conditions
  • 4. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 4 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
  • 5. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 5 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
  • 6. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 6 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
  • 7. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 7 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
  • 8. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 8 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hamiltonian Cycle Definition (Hamiltonian Cycle) A Hamiltonian cycle is a cycle in a graph passing through all the vertices once.
  • 9. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 9 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 10. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 10 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 11. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 11 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 12. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 12 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 13. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 13 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 14. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 14 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 15. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 15 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Tour Cost: Hamiltonian Cycle Cost
  • 16. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 16 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveling Salesman Problem (TSP) Definition INPUT: Edge weighted Graph G = (V,E) OUTPUT: Minimum cost Hamiltonian Cycle.
  • 17. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente 17 Introduction Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveling Salesman Problem (TSP) Definition INPUT: Edge weighted Graph G = (V,E) OUTPUT: Minimum cost Hamiltonian Cycle.
  • 18. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 18 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Optimization Problems [Papadimitriou and Steiglitz, 1998] Definition (Instance of an Optimization Problem) An instance of an optimization problem is a pair (F, c), where F is any set, the domain of feasible points; c is the cost function, a mapping c : F ! R The problem is to find an f 2 F for which c(f ) c(y) 8 y 2 F Such a point f is called a globally optimal solution to the given instance, or, when no confusion can arise, simply an optimal solution.
  • 19. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 18 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Optimization Problems [Papadimitriou and Steiglitz, 1998] Definition (Instance of an Optimization Problem) An instance of an optimization problem is a pair (F, c), where F is any set, the domain of feasible points; c is the cost function, a mapping c : F ! R The problem is to find an f 2 F for which c(f ) c(y) 8 y 2 F Such a point f is called a globally optimal solution to the given instance, or, when no confusion can arise, simply an optimal solution. Definition (Optimization Problem) An optimization problem is a set of instances of an optimization problem.
  • 20. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 19 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Optimization Problems [Papadimitriou and Steiglitz, 1998] Two categories 1. with continuous variables 2. with discrete variables, which we call combinatorial
  • 21. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 20 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Combinatorial Optimization Problem [Papadimitriou and Steiglitz, 1998] Definition (Combinatorial Optimization Problem ) An optimization problem consists of = (D,R, cost, goal) 1. D: A set of all valid input instances. Input instance I 2 D 2. R: A set of all feasible solution. Each I 2 D has a set of feasible solutions, R(I ). 3. cost: Objective function that assigns a non-negative rational number to each pair (I , SOL) cost(I , SOL), where I 2 D and SOL 2 R(I ). 4. goal 2 {min, max}.
  • 22. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 21 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
  • 23. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 22 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: As a Graph Problem
  • 26. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction 25 Optimization Problems Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Growth of N!
  • 27. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable!
  • 28. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable! TSP is NP-hard!
  • 29. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable! TSP is NP-hard! Unless P=NP, we cannot obtain an efficient algorithm solving the problem.
  • 30. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 26 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Intractable Problems TSP is Intractable! TSP is NP-hard! Unless P=NP, we cannot obtain an efficient algorithm solving the problem.
  • 31. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 27 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity NP-hard Problems Problem A is NP-hard if all problems in NP are polynomial-time reducible to it. NP-hard problems are at least as hard as the hardest problems in NP.
  • 32. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 28 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes
  • 33. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 28 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes
  • 34. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 28 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes I P is the class of decision problems solvable in polynomial time by a Turing machine (TM). I NP is the class of decision problems verifiable in polynomial time by a Turing Machine. I A decision problem is NP-complete if (1) it is in NP, and (2) any problem in NP can be reduced to it (under some notion of reduction). The class of NP-complete problems is sometimes called NPC.
  • 35. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 29 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Examples of NP-hard Problems [Garey and Johnson, 1979] 1. Satisfiability problem 2. Clique Problem 3. Independent set problem 4. Vertex Cover Problem 5. Traveling salesman problem 6. Median String Problem 7. and many more
  • 36. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 30 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems Design technique have a well specified structure that even provides a framework for possible implementations Concepts formulate ideas and rough frameworks about how to attack hard algorithmic problems .
  • 37. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 30 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems Design technique have a well specified structure that even provides a framework for possible implementations I Dynamic Programming I Branch and Bound I Greedy Schema I Genetic Algorithm I Simulated Annealing, Tabu Search I Local search I . . . Concepts formulate ideas and rough frameworks about how to attack hard algorithmic problems . I Approximation Algorithms I Parameterized Algorithms I Randomized Algorithms I . . .
  • 38. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 31 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems I Exact algorithm always obtain the optimal solution I Approximation algorithm settle for good enough solutions. Goodness of solution is guaranteed and measured using approximation ratio. I Heuristic algorithms produce solutions, which are not guaranteed to be close to the optimum. The performance of heuristics is often evaluated empirically.
  • 39. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 32 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approaches in Solving NP-hard Problems I Exact algorithm always obtain the optimal solution I Approximation algorithm settle for good enough solutions. Goodness of solution is guaranteed and measured using approximation ratio. I Heuristic algorithms produce solutions, which are not guaranteed to be close to the optimum. The performance of heuristics is often evaluated empirically.
  • 40. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 33 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Triangle inequality Edge weighted graph with cost function c satisfies triangle inequality if for all u, v, and w 2 V c(u, w) c(u, v) + c(v, w) .
  • 41. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 34 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Triangle inequality c(u, w) c(u, v) + c(v, w)
  • 42. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 35 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Solving Traveller’s Dilemma INPUT: Edge weighted graph 1. Compute a Minimum Spanning Tree for G 2. Select a vertex r 2 V(G) to be the root vertex 3. Let L be the list of vertices visited in a preorder walk OUTPUT: Hamiltonian Cycle that visits A preorder tree walk recursively visits every vertex in the tree, listing a vertex when it is first encountered , before any of its children are visited.
  • 43. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 36 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 44. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 37 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 45. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 38 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 46. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 39 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 47. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 40 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 48. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 41 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 49. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 42 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 50. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 43 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 51. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 44 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 52. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 45 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 53. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 46 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Computing the Minimum Spanning Tree
  • 54. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 47 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 55. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 48 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 56. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 49 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 57. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 50 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 58. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 51 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 59. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 52 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 60. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 53 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Preorder
  • 61. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 54 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hamiltonian Cycle from the Preorder Walk
  • 62. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 55 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hamiltonian Cycle from the Preorder Walk 4 + 4 + 10 + 15 + 10 + 16 = 59
  • 63. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 56 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Total tour cost vs. the optimal cost Let H denote an optimal Hamiltonian tour in G. Let T be the minimum spanning tree of G. c(T) c(H) Let W be the full length cost of the preorder walk. c(W) = 2c(T) Let H be the output Hamiltonian Cycle from the algorithm c(W) c(H) Then c(H) 2c(H)
  • 64. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems 56 Hard Problems Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Total tour cost vs. the optimal cost Let H denote an optimal Hamiltonian tour in G. Let T be the minimum spanning tree of G. c(T) c(H) Let W be the full length cost of the preorder walk. c(W) = 2c(T) Let H be the output Hamiltonian Cycle from the algorithm c(W) c(H) Then c(H) 2c(H) Approximation ratio () is 2
  • 65. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 57 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Algorithms Definition (Approximation Algorithm [Williamson and Shmoy, 2010] ) An -approximation algorithm for an optimization problem is a polynomial-time algorithm that for all instances of the problem produces a solution whose value is within a factor of of the value of an optimal solution. Given a problem instance I with an optimal solution Opt(I ), i.e. the cost function cost(Opt(I )) is minimum/maximum.
  • 66. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 57 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Algorithms Definition (Approximation Algorithm [Williamson and Shmoy, 2010] ) An -approximation algorithm for an optimization problem is a polynomial-time algorithm that for all instances of the problem produces a solution whose value is within a factor of of the value of an optimal solution. Given a problem instance I with an optimal solution Opt(I ), i.e. the cost function cost(Opt(I )) is minimum/maximum. I An algorithm for a minimization problem is called -approximative algorithm for some 1, if the algorithm obtains a maximum cost of · cost(Opt(I )), for any input instance I .
  • 67. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 57 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Algorithms Definition (Approximation Algorithm [Williamson and Shmoy, 2010] ) An -approximation algorithm for an optimization problem is a polynomial-time algorithm that for all instances of the problem produces a solution whose value is within a factor of of the value of an optimal solution. Given a problem instance I with an optimal solution Opt(I ), i.e. the cost function cost(Opt(I )) is minimum/maximum. I An algorithm for a minimization problem is called -approximative algorithm for some 1, if the algorithm obtains a maximum cost of · cost(Opt(I )), for any input instance I . I An algorithm for a maximization problem is called -approximative algorithm, for some 1, if the algorithm obtains a minimum cost of · cost(Opt(I )), for any input instance I .
  • 68. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 58 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio I Minimization, 1 cost(Opt(I )) cost(SOL) cost(Opt(I )) I Maximization, 1 cost(Opt(I )) cost(SOL) cost(Opt(I ))
  • 69. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 59 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio I Additive Approximation Algorithms: SOL OPT + c I Constant Approximation Algorithms: SOL c · OPT I Logarithmic Approximation Algorithms: SOL = O(log n) · OPT I Polynomial Approximation Algorithms: SOL = O(nc) · OPT, where c 1
  • 70. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 60 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximable Problems [Vazirani, 2001] Definition (APX) I An abbreviation for “Approximable, is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant.
  • 71. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 60 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximable Problems [Vazirani, 2001] Definition (APX) I An abbreviation for “Approximable, is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant. I Problems in this class have efficient algorithms that can find an answer within some fixed percentage of the optimal answer.
  • 72. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 61 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Polynomial-time Approximation Schemes Definition (PTAS [Aaronson et al., 2008]) The subclass of NPO problems that admit an approximation scheme in the following sense. For any 0, there is a polynomial-time algorithm that is guaranteed to find a solution whose cost is within a 1 + factor of the optimum cost. Contains FPTAS, and is contained in APX.
  • 73. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 61 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Polynomial-time Approximation Schemes Definition (PTAS [Aaronson et al., 2008]) The subclass of NPO problems that admit an approximation scheme in the following sense. For any 0, there is a polynomial-time algorithm that is guaranteed to find a solution whose cost is within a 1 + factor of the optimum cost. Contains FPTAS, and is contained in APX. Definition (FPTAS [Aaronson et al., 2008] ) The subclass of NPO problems that admit an approximation scheme in the following sense. For any 0, there is an algorithm that is guaranteed to find a solution whose cost is within a 1 + factor of the optimum cost. Furthermore, the running time of the algorithm is polynomial in n (the size of the problem) and in 1/.
  • 74. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 62 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio of well known Hard Problems 1. FPTAS: Bin Packing Problem 2. PTAS: Makespan Scheduling Problem 3. APX: 3.1 Min Steiner Tree Problem (1.55 Approximable) [Robins,2005] 3.2 Min Metric TSP (3/2 Approximable) [Christofides,1977] 3.3 Max SAT (0.77 Approximable) [Asano,1997] 3.4 Vertex Cover (2 Approximable) [Vazirani,2001] 4. MAX SNP 4.1 Independent Set Problem 4.2 Clique Problem 4.3 Travelling Salesman Problem
  • 75. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 63 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Complexity Classes [Ausiello et al., 2011] FPTAS ( PTAS ( APX ( NPO
  • 76. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 64 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Inapproximable Problems Many problems have polynomial-time approximation schemes. However, there exists a class of problems that is not so easy [Williamson and Shmoy, 2010]. This class is called MAX SNP. Theorem For any MAXSNP-hard problem, there does not exist a polynomial-time approximation scheme, unless P = NP [Williamson and Shmoy, 2010].
  • 77. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 64 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Inapproximable Problems Many problems have polynomial-time approximation schemes. However, there exists a class of problems that is not so easy [Williamson and Shmoy, 2010]. This class is called MAX SNP. Theorem For any MAXSNP-hard problem, there does not exist a polynomial-time approximation scheme, unless P = NP [Williamson and Shmoy, 2010].
  • 78. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 64 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Inapproximable Problems Many problems have polynomial-time approximation schemes. However, there exists a class of problems that is not so easy [Williamson and Shmoy, 2010]. This class is called MAX SNP. Theorem For any MAXSNP-hard problem, there does not exist a polynomial-time approximation scheme, unless P = NP [Williamson and Shmoy, 2010]. Theorem If P6= NP, then for any constant 1, there is no polynomial-time approximation algorithm with approximation ratio for the general travelling salesman problem.
  • 79. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 65 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: with additional informa-tion
  • 80. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 66 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: with additional informa-tion
  • 81. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 67 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: with additional informa-tion
  • 82. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 68 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009]
  • 83. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 69 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009]
  • 84. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 70 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009]
  • 85. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 71 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Traveller’s Dilemma: Nearest Insert [Ausiello et al., 2009] 10 + 12 + 6 + 4 + 5 + 11 = 48
  • 86. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 72 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Comparison
  • 87. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 72 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Comparison 10+12+6+4+5+11 = 48
  • 88. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 72 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Comparison 10+12+6+4+5+11 = 48 4+4+10+15+10+16 = 59
  • 89. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 73 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1
  • 90. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 73 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Approximation Ratio Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1 Theorem There is a 3/4 Approximation algorithm for solving Metric TSP.
  • 91. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 74 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity 3/2- Approximation Algorithm [Ausiello et al., 2009] Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1 Algorithm: Output the best solution between (H1,H2), where H1 =Nearest Insert(In,H n , In+1) H2 = 32 -Approximation Algorithm(In+1)
  • 92. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 74 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity 3/2- Approximation Algorithm [Ausiello et al., 2009] Definition (Metric TSP with additional info) INPUT: In,H n , In+1 OUTPUT: Hn+1 Algorithm: Output the best solution between (H1,H2), where H1 =Nearest Insert(In,H n , In+1) H2 = 32 -Approximation Algorithm(In+1) Proof. n+1) + 2d(v, n + 1) Triangle Inequality ) c(H1) c(H c(H1) c(H n+1) + 2max(d(i, n + 1), d(j, n + 1))) c(H2) 3/2c(H n+1) − max(d(i, n + 1), d(j, n + 1))) min(c(H1), c(H2)) (1/3)c(H1) + (2/3)c(H2) 4/3c(H n+1)
  • 93. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems 75 Approximation Algorithms Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization [Ausiello et al., 2009] The concept of reoptimization can in many cases help to improve the approximability of hard optimization problems by lowering constant approximation ratios or even constructing a PTAS for problems which are APX-hard in their classical non reoptimization variant.
  • 94. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 76 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Don’t start from scratch when confronted with a problem, but try to make good use of prior knowledge about similar problem instances whenever they are available.
  • 95. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 76 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Don’t start from scratch when confronted with a problem, but try to make good use of prior knowledge about similar problem instances whenever they are available. Definition (Reoptimization [Hans-joachim Böckenhauer, 2008] ) Given a problem instance and an optimal solution for it, we are to efficiently obtain an optimal solution for a locally modified instance of the problem.
  • 96. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 76 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Don’t start from scratch when confronted with a problem, but try to make good use of prior knowledge about similar problem instances whenever they are available. Definition (Reoptimization [Hans-joachim Böckenhauer, 2008] ) Given a problem instance and an optimal solution for it, we are to efficiently obtain an optimal solution for a locally modified instance of the problem. Definition (Postoptimality Analysis [Hoesel and Wagelmans, 1999]) Postoptimality analysis deals with the related question of how much an instance may be altered without changing the set of optimal solutions.
  • 97. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 77 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
  • 98. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 78 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
  • 99. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 79 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
  • 100. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 80 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization: Minimum Spanning Tree Example
  • 101. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 81 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization Definition (Reoptimization [Zych, 2012]) Let = (D,R, cost, goal) be an optimization problem and M D × D be a binary relation (the modification). The corresponding reoptimization problem RM() = (DRM(),RRM(), costRM(), goalRM()) consists of 1. a set of feasible instances defined as DRM() = {(I , I 0, SOL) : (I , I 0) 2M and SOL 2 R(I )}; we refer to I as the original instance and to I 0 as the modified instance 2. a feasibility relation defined as RRM()((I , I 0, SOL)) = R(I 0)
  • 102. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 82 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hardness of Reoptimization I NP hardness of a problem implies that the solution to a locally modified problem cannot be found in polynomial-time, given the solution of the original problem. I Reoptimization of a Hard Problem is still NP-hard! [Hans-joachim Böckenhauer, 2008]. Proof. Let be NP-hard. Let RM() be the reoptimization variant of To prove that RM() is NP-hard, we need to show that we can reduce to RM().
  • 103. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 82 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Hardness of Reoptimization I NP hardness of a problem implies that the solution to a locally modified problem cannot be found in polynomial-time, given the solution of the original problem. I Reoptimization of a Hard Problem is still NP-hard! [Hans-joachim Böckenhauer, 2008]. Proof. Let be NP-hard. Let RM() be the reoptimization variant of To prove that RM() is NP-hard, we need to show that we can reduce to RM(). I But it is an advantage to know a solution of a problem instance, as compared to not knowing it.
  • 104. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 83 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization 1. Reoptimization can help in providing efficient algorithms to problems involved in dynamic systems 2. Reoptimization can help in providing a better solution or an efficient algorithm for computationally hard problems
  • 105. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 83 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Reoptimization 1. Reoptimization can help in providing efficient algorithms to problems involved in dynamic systems I Shortest Path Problem [Pallottino and Scutella, 2003] [Nardelli et al., 2003] I Dynamic Minimum Spanning Tree [Thorup, 2000] with edge weights [Ribeiro and Toso, 2007] [Cattaneo et al., 2010] I Vehicle Routing Problem [Secomandi and Margot, 2009] I Facility Location Problem [Shachnai et al., 2012] 2. Reoptimization can help in providing a better solution or an efficient algorithm for computationally hard problems I Traveling salesman problem [Královic and Mömke, 2007] [Hans-joachim Böckenhauer, 2008] [Ausiello et al., 2011] I Steiner tree problem [Hromkovic, 2009] [Zych and Bilò, 2011][Bilo and Zych, 2012] [Böckenhauer et al., 2012] I Shortest common superstring [Bilo,2011] [Popov, 2013] I Hereditary problems on Graphs [Boria et al., 2012a] I Scheduling Problem [Boria et al., 2012b] I Pattern Matching [Yue and Tang, 2008] [?]
  • 107. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 85 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems Definition (Self-reducibility [Zych, 2012]) We will say that a problem is self-reducible if there is a polynomial-time algorithm, , satisfying the following conditions. 1. Given an instance I and an atom of a solution to I , outputs an instance I. We require that the size of I is smaller than the size of I , i.e. |I| |I |. Let R(I |) represent the set of feasible solutions to I containing the atom . We require that every solution SOL of I, i.e., SOL 2 R(I), has a corresponding SOL [ {} 2 R(I |) and that this correspondence is one-to-one. 2. For any set H 2 R(I) it holds that the cost(I ,H [ {}) = cost(I , ) + cost(I,H).
  • 108. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 86 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] An atom, as used in this context, constitutes a solution for a specific problem instance. For example, in a maximum independent set problem, atoms are the vertices of the graph which constitutes the solution.
  • 109. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 87 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Example Weighted Maximum Independent Set is self-reducible G = (V,E) SOL 2 R(G) () (SOL V(G)) and SOL is independent in G Atoms(G) = V(G) Let be the self-reduction function Gv = (G, v)
  • 110. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 87 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Example Weighted Maximum Independent Set is self-reducible G = (V,E) SOL 2 R(G) () (SOL V(G)) and SOL is independent in G Atoms(G) = V(G) Let be the self-reduction function Gv = (G, v)
  • 111. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 87 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Example Weighted Maximum Independent Set is self-reducible G = (V,E) SOL 2 R(G) () (SOL V(G)) and SOL is independent in G Atoms(G) = V(G) Let be the self-reduction function Gv = (G, v) (1) SOL = SOL [ {v}. (2) cost(G, SOL) = cost(Gv, SOLv) + c(v)
  • 112. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 88 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Self-reducible Problems [Zych, 2012] Definition (Self-reducible problems) Set of combinatorial problems which admits the self-reducibility property (e.g., Max SAT, Min Steiner Tree) Classified different types of modifications 1. Progressing Subtractive 2. Progressing Additive 3. Regressing Subtractive 4. Regressing Additive Table: Summary of approximation ratio from [Zych, 2012]
  • 113. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 89 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012]
  • 114. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 90 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012]
  • 115. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 91 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012]
  • 116. 103 Reoptimization Techniques for Solving Hard Problems Jhoirene B Clemente Introduction Optimization Problems Hard Problems Approximation Algorithms 92 Reoptimization Other Results on Reoptimization Heredity Problems References Algorithms and Complexity Types of Modifications [Zych, 2012] Table: Summary of approximation ratio from [Zych, 2012]
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