This document presents an approximation algorithm for the vertex cover problem. It first defines approximation algorithms as techniques for dealing with NP-complete optimization problems that provide near-optimal solutions in polynomial time. It then defines the vertex cover problem as finding a subset of vertices that covers all graph edges, and provides an example 2-approximation algorithm that iteratively adds vertices covering arbitrary remaining edges to the cover set.

1. Presentation of Design Analysis and Algorithm on
Approximation Algorithm and
Vertex cover problem
Presented By: Sumit Gyawali
Roll no:325
Bsc csit/3rd year/5th sem
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2. overview
 Approximation Algorithm
 Vertex cover problem
 Example
 vertex cover Algorithm
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3. Approximation Algorithms
 It is a way of dealing with NP-completeness for optimization problem.
 This technique does not guarantee the best solution.
 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 most polynomial time.
 If we are dealing with optimization problem (maximization or minimization)
with feasible solution having positive cost then it is worthy to look at
approximate algorithm for near optimal solution.
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4. Continue..
 An algorithm has an approximate ratio of ρ(n) if, for any problem of input size
n, the cost C of solution by an algorithm and the cost C* of optimal solution
have the relation as max(C/C*,C*,C) ≤ ρ(n). Such an algorithm is called ρ(n)-
approximation algorithm.
 The relation applies for both maximization (0 < C ≤ C*) and minimization (0 <
C* ≤ C) problems. ρ(n) is always greater than or equal to 1. If solution
produced by approximation algorithm is true optimal solution then clearly we
have ρ(n) = 1.
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5. Vertex cover Problem
 A vertex cover of an undirected graph is a subset of its vertices such that for
every edge (u, v) of the graph, either ‘u’ or ‘v’ is in vertex cover. Although the
name is Vertex Cover, the set covers all edges of the given graph.
 It is also known as NP Complete problem, i.e., there is no polynomial time
solution for this unless P = NP. There are approximate polynomial time
algorithms to solve the problem though.
Following are some Example:
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6. Vertex cover Problem
Algorithm:
ApproxVertexCover (G)
{
C { } ;
E’ = E
while E` is not empty
do Let (u, v) be an arbitrary edge of E`
C = C ≈ {u, v}
Remove from E` every edge incident on either u or v
return C
}
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7. Reference
www.bsccsitnotes.com
www.google.com
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