Approximation algorithms

Approximation
Algorithms
Bipesh Subedi
ME 2021, Computer Engineering
DoCSE

Outline
2
➔ Introduction
➔ Optimization Problem
➔ Statement
➔ Approximation Ratio
➔ Examples of Approximation algorithms
➔ Travelling Salesman Problem (TSP)
➔ Analysis
➔ Application
➔ Pros and Cons
➔ ‘Conclusion

Introduction
Approximation algorithms refers to the efficient algorithms that finds a
solution to an optimization problems ( in particular NP-Hard problems )
that is provably close to optimal solution.
When it is used ?
➔ When finding an optimal solution is very difficult
➔ In some cases, where exact solution is not needed and near-
optimal solution can be found quickly
3

➔ For NP-Complete problems, there is possibly no
polynomial-time algorithms to find an optimal solution.
➔ The idea of approximation algorithms is to develop
polynomial-time algorithms to get near optimal solution.
4

Optimization Problem
It refers to the problem of finding best option/solution among all
feasible/possible solutions.
Various important applied problems involve in finding the best possible way
to accomplish the task which is often done by finding the maximum or
minimum values to a function.
➔ Maximization Problem : Finding the maximum optimal value
➔ Minimization Problem : Finding the minimal optimum value
For example: the minimum time to make a certain journey, the minimum
cost for doing a task, the maximum power that can be generated by a
device, and so on.
5

Statement
➔ An algorithm for a problem of size n has an approximation ratio
ρ(n) if for any input, the algorithm produces a solution with cost
C that satisfies the property:
Where Copt is the cost of optimal solution and ρ(n) is a constant or a
function of n.
6

Approximation Ratio
➔ The approximation ratio of an algorithm is the ratio
between the result obtained by the algorithm and the
optimal cost.
Significance
➔ Performance Guarantee:
It guarantees that the result produced by the approximation
algorithm is not going to be worse than a factor of ρ(n)
compared to the optimal solution.
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The algorithm with approximation ratio ρ(n) is then called
ρ(n)-approximation algorithm.
Here we will consider our approximation algorithm to have
constant approximation ratio.
For example: If ρ(n) = 2 then,
For maximization problem : our solution will be no less than half
the optimal solution. ( Copt/C <= 2 )
For minimization problem : our solution will be no more than
twice the optimal solution. (C/Copt <= 2)
8

Range of Approximation ratio
➔ If ρ(n) =1 then,
The algorithm can always find a
optimal solution
➔ If ρ(n) >1 then,
It gives the approximation in which
the algorithm works.
9

10
Examples of Approximation algorithms
➔ Travelling salesman Problem
➔ Vertex-Cover Problem
➔ Bin Packing

Travelling Salesman Problem (TSP)
A salesman should visit all the cities exactly once starting from one
city and return back to the starting city such that the total length of the
tour should be minimum.
➔ It is a NP-Complete problem so, there is no polynomial time.
➔ Use an approximation algorithm with a constant approximation
ratio.
11

Special Case of
Travelling
Salesman
Problem (TSP)
Triangle Inequality Property
a + b >= c
b + c >= a
a + c >= b
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Approach
13
Step 1: Start with a complete graph.
Step 2: Compute a Minimum Spanning
Tree (MST) using algorithms like Prim’s,
kruskal’s.
Step 3: Perform a Depth-First Search
(DFS) traversal.
Step 4: Delete duplicate vertices.
Step 5: Join the edges in the order to
obtain approx. tour walk.

Given a weighted, undirected graph,
start from certain vertex, find a
minimum route to visit each vertices
once, and return to the original
vertex.
14

Computing Minimum Spanning Tree
Kruskal’s algorithm
Steps:
1. Sort all the edges in non-decreasing order
of their weight.
2. Pick the smallest edge. Check if it forms a
cycle in the spanning tree formed so far. If no
cycle is formed, include this edge. Otherwise,
discard it.
3. Repeat step 2 until it is a spanning tree.
15

After computing the
minimum spanning tree ,
we perform a depth-first
search in the MST
starting from a vertex A.
16

18
A -> B -> D -> B -> E -> B -> A

19
A -> B -> D -> B -> E -> B -> A -> C -> A

After deleting duplicate vertices we
get, A -> B -> D -> E -> C -> A path.
Joining the edges in the above
order we obtain the approx. tour
route.
20

Analysis
➔ Depth first tree tour (DFTT) length is exactly twice the MST’s weight.
➔ MST weight is not more than the length of optimal tour
➔ The tour length is at most twice the optimal length i.e C <= 2 x Copt
So, in this case ρ(n) = 2
Cost found by the APPROX-MST-DFS algorithm :
C = 15 + 10 + 25 + 25 + 30 = 105
Optimal cost ( Copt ) = Sum of MST weights = 15 + 10 + 25 + 10 = 60
C/Copt <= ρ(n)
105/60 <= 2
1.75 <= 2
21

➔ For companies like FedEx and their competitors traveling
salesman problem is used as a reference to solve related
/similar problems.
➔ Can be used in network design and routing purposes.
➔ Used in astronomy to determine the fastest way of drilling an
aluminium disk placed at the focal point of telescope for each
spot of interest to collect the light from the galaxy or star over
a given period of time .
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Application

Pros
➔ Deals with NP- Complete
problems
➔ Finds solutions which are close
to optimal solution in
polynomial time
➔ Cost Effective
Cons
➔ This technique does not
guarantee the best solution
➔ May not be useful in all
practical application
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Conclusion
To conclude we can say that the goal of an approximation
algorithm is to come as close as possible to the optimum value
in a reasonable amount of time which is at the most polynomial
time.
24

Approximation algorithms

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