2-Approximation Vertex Cover

ApproximationAlgorithm:

TheVertex-coverProblem

Sudipta SahaShubha1205014

Repon KumarRoy 1205002

MotivationalProblem

Approximation Algorithm: The Vertex-coverProblem

Sudipta Saha Shubha 1205014Repon Kumar Roy 1205002

Department of Computer Science and EngineeringBangladesh University of Engineering and Technology

December 24, 2015





MotivationalProblem

Table of Contents

1 Motivational Problem





MotivationalProblem

Motivational Problem : Modeling in Graph





MotivationalProblem

Formal Definition of Vertex Cover Problem

A vertex cover of an undirected graph

G = (V ,E ) is a subset V ′ ∈ V such that

if (u, v) is an edge of G , then either

u ∈ V ′ or v ∈ V ′(or both).

The vertex-cover problem is to find a

vertex cover of minimum size in a given

undirected graph. We call such a vertex

cover an optimal vertex cover.





MotivationalProblem














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MotivationalProblem

A Solution may be . . .





Way to Solution . . .

Time Complexity is O(2n × n)Time Complexity is exponential oninputNP-complete problem






Time Complexity is O(2n × n)

Time Complexity is exponential oninputNP-complete problem






Time Complexity is O(2n × n)Time Complexity is exponential oninput

NP-complete problem





TheApproximateAlgorithm

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Time Complexity is O(2n × n)Time Complexity is exponential oninputNP-complete problem






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A language L ⊆ {0, 1}∗ isNP-complete if1. L ∈ NP , and2. L′ ≤p L for every L′ ∈ NP .






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Can be solved using dynamicprogramming in polynomial timewhen input graph is a tree.






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Needs approximation algorithm forgeneral graphWe call an algorithm that returnsnear-optimal solutions anapproximation algorithm






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Needs approximation algorithm forgeneral graph

We call an algorithm that returnsnear-optimal solutions anapproximation algorithm






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Needs approximation algorithm forgeneral graphWe call an algorithm that returnsnear-optimal solutions anapproximation algorithm






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Table of Contents

2 The Approximate Algorithm

3 Applications

4 Some Questions






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Simulation






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Simulation






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Simulation






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Simulation






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Simulation






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Time Complexity

O(V + E )






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Proof of 2-Approximation

We say that an algorithm for a problem has anapproximation ratio of ρ(n) if, for any input ofsize n, the cost X of the solution produced bythe algorithm is within a factor of ρ(n) of thecost X ∗ of an optimal solution:

max( XX ∗ ,

X ∗

X )≤ ρ(n).






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Proof of 2-Approximation(Is it a Vertex-Cover?)






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We need to prove that |X | ≤ 2|X ∗|We will first prove that |X ∗| ≥ |Y |






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We need to prove that |X | ≤ 2|X ∗|

We will first prove that |X ∗| ≥ |Y |






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We need to prove that |X | ≤ 2|X ∗|We will first prove that |X ∗| ≥ |Y |






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Proof of 2-Approximation(|X ∗| ≥ |Y |)






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Proof of 2-Approximation(|X ∗| ≥ |Y |) : AnotherSubset of Arbitrary Edges






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Now we will prove that, |X | = 2|Y |






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Proof of 2-Approximation(|X | = 2|Y |)






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|X | = 2|Y ||Y | ≤ |X ∗|So,|X | ≤ 2|X ∗|thereby proving 2-Approximation






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Table of Contents


3 Applications

4 Some Questions






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Selecting Optimal Position of Security

Cameras.

Setting of ATM booths Optimally.

For Bipartitle Graph, Maximum

Matiching = Minimum Vertex Cover.






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Cameras.









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Cameras.









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Cameras.









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Table of Contents


3 Applications

4 Some Questions






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Is there any graph where this

approximate algorithm always gives

twice the size of optimal solution?

How about repeatedly selecting a

vertex of highest degree, and removing

all of its incident edges?






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Thanks and Questions are Welcome

2-Approximation Vertex Cover

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