NP-Complete Problem

NP-COMPLETE
Prof. Manjusha Amritkar
Assistant Professor
Department of Information Technology
Hope Foundation’s
International Institute of Information
Technology, I²IT
www.isquareit.edu.in

THE TRAVELING SALESMAN PROBLEM
 Suppose that you are given the road map of India.
 You need to find a traversal that covers all the
cities/towns/villages of population ≥ 1, 000.
 And the traversal should have a short distance, say,
≤ 9, 000 kms.
 You will have to generate a very large number of
traversals to find out a short traversal.
 Suppose that you are also given a claimed short
traversal.
 It is now easy to verify that given claimed traversal
is indeed a short traversal.
Hope Foundation’s International Institute of Information Technology, I²IT P-14,Rajiv Gandhi Infotech Park
MIDC Phase 1, Hinjawadi, Pune – 411057 Tel - +91 20 22933441/2/3 | www.isquareit.edu.in | info@isquareit.edu.in

THE BIN PACKING PROBLEM
 Suppose you have a large container of volume 1000
cubic meter and 150 boxes of varying sizes with
volumes between 10 to 25 cubic meters.
 You need to fit at least half of these boxes in the
container.
 You will need to try out various combinations of 75
boxes (there are 1040 combinations) and various ways
of laying them in the container to find a fitting.
 Suppose that you are also given a set of 75 boxes and
a way of laying them.
 It is now easy to verify if these 75 boxes layed out in
the given way will fit in the container.

HALL-I ROOM ALLOCATION
 Each wing of Hall-I has 72 rooms.
 Suppose from a batch of 540 students, 72 need to be
housed in C-wing.
 There are several students that are “incompatible”
with each other, and so no such pair should be
present in the wing.
 If there are a large number of incompatibilities, you
will need to try out many combinations to get a
correct one.
 Suppose you are also given the names of 72
students to be housed.
 It is now easy to verify if they are all compatible.

DISCOVERY VERSUS VERIFICATION
 In all these problems, finding a solution appears to
be far more difficult than checking the correctness of
a given solution.
 Informally, this makes sense as discovering a
solution is often much more difficult than verifying its
correctness.
 Can we formally prove this?
 Leads to the P versus NP problem.

FORMALIZING EASY-TO-SOLVE
 A problem is easy to solve if the solution can be
computed quickly.
 Gives rise to two questions:
I. How is it computed?
II. How do we define “quickly”?

COMPUTING METHOD
 We will use an algorithm to compute.
 In practice, the algorithm will run on a computer via
a computer program.

ALGORITHMS
 An algorithm is a set of precise instructions for
computation.
 The algorithm can perform usual computational
steps, e.g., assignments, arithmetic and Boolean
operations, loops.
 For us, an algorithm will always have input presented
as a sequence of bits.
 The input size is the number of bits in the input to the
algorithm.
 The algorithm stops after outputting the solution.

TIME COMPLEXITY OF PROBLEMS
 A problem has time complexity TA(n) if there is an
algorithm A that solves the problem on every input.
 Addition has time complexity O(n).
 Multiplication has time complexity O(n2)

TIME MEASUREMENT
 Let A be an algorithm and x be an input to it.
 Let TA(x) denote the number of steps of the algorithm
on input x.
 Let TA(n) denote the maximum of TA(x) over all inputs
x of size n.
 We will use TA(n) to quantify the time taken by
algorithm A to solve a problem on different input sizes.
 For example, an algorithm A that adds two n bit
numbers using school method has TA(n) = O(n).
 An algorithm B that multiplies two n bits numbers
using school method has TA(n) = O(n2)

QUANTIFYING EASY-TO-COMPUTE
 The problems of adding and multiplying are definitely
easy.
 Also, if a problem is easy, and another problem can be
solved in time n · T(n) where T(n) is the time
complexity of the easy problem, then the new problem
is also easy.
 This leads to the following definition:
A problem is efficiently solvable if its time complexity is n
O(1) . Such problems are also called polynomial-time
problems.

THE CLASS P
 The class P contains all efficiently solvable
problems.
 Specifically, they are the problems that can be
solved in time O(n k) for some constant k, where n
is the size of input to the problem.

THE CLASS NP
 NP = Non-Deterministic polynomial time
 The class NP contains those problems that are
“verifiable” in polynomial time.
 e.g 1. Hamiltonian cycle

HAMILTONIAN CYCLE
 Determining whether a directed graph has a
Hamiltonian cycle does not have a polynomial time
algorithm (yet!)
 However if someone was to give you a sequence of
vertices, determining whether or not that sequence
forms a Hamiltonian cycle can be done in
polynomial time
 Therefore Hamiltonian cycles are in NP

SAT
 A boolean formula is satisfiable if there exists
some assignment of the values 0 and 1 to its
variables that causes it to evaluate
to 1.
 CNF – Conjunctive Normal Form. ANDing of
clauses of ORs

2-CNF SAT
 Each or operation has two arguments that are either
variables or negation of variables
 The problem in 2 CNF SAT is to find true/false(0 or 1)
assignments to the variables in order to make the entire
formula true.
 Any of the OR clauses can be converted to implication
clauses
(xy)(yz)(xz)(zy)

2-SAT IS IN P
 Create the implication graph
x
y
x
z
z
y

SATISFIABILITY VIA PATH FINDING
 If there is a path from
 And if there is a path from
 Then FAIL!
 How to find paths in graphs?
 DFS/BFS and modifications thereof

3 CNF SAT (3 SAT)
 Not so easy anymore.
 Implication graph cannot be constructed
 No known polytime algorithm
 Is it NP?
 If someone gives you a solution how long does it take to
verify it?
 Make one pass through the formula and check
 This is an NP problem

REFERENCES
 Contents are referred from following web
resources
 https://www.seas.upenn.edu/~bhusnur4/cit596_spri
ng2014/PNP.pptx
 https://www.cse.iitk.ac.in/users/manindra/presentati
ons/IITKTalk.pdf

THANK YOU
For further details, please contact
Manjusha Amritkar
manjushaa@isquareit.edu.in
Department of Information Technology
Hope Foundation’s
International Institute of Information Technology, I²IT
P-14,Rajiv Gandhi Infotech Park
MIDC Phase 1, Hinjawadi, Pune – 411057
Tel - +91 20 22933441/2/3
www.isquareit.edu.in | info@isquareit.edu.in

NP-Complete Problem

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