2-Approximation Vertex Cover
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This document presents an approximation algorithm for the vertex cover problem in graphs. It begins with definitions of the vertex cover problem and shows that finding an optimal solution is NP-complete. It then presents a 2-approximation algorithm that finds a vertex cover of size at most twice the optimal. The time complexity of the algorithm is O(V+E). Applications of the vertex cover problem and some open questions are also discussed.
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2-Approximation Vertex Cover
- 1. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Department of Computer Science and Engineering Bangladesh University of Engineering and Technology December 24, 2015
- 2. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem Table of Contents 1 Motivational Problem
- 3. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem Motivational Problem : Modeling in Graph
- 4. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem Formal Definition of Vertex Cover Problem
- 5. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem 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).
- 6. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem 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.
- 7. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Motivational Problem A Solution may be . . .
- 8. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Way to Solution . . .
- 9. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Way to Solution . . . Time Complexity is O(2n × n)
- 10. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 Way to Solution . . . Time Complexity is O(2n × n) Time Complexity is exponential on input
- 11. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Way to Solution . . . Time Complexity is O(2n × n) Time Complexity is exponential on input NP-complete problem
- 12. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Way to Solution . . . A language L ⊆ {0, 1}∗ is NP-complete if 1. L ∈ NP, and 2. L ≤p L for every L ∈ NP.
- 13. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Way to Solution . . . Can be solved using dynamic programming in polynomial time when input graph is a tree.
- 14. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Way to Solution . . .
- 15. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Way to Solution . . . Needs approximation algorithm for general graph
- 16. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Way to Solution . . . Needs approximation algorithm for general graph We call an algorithm that returns near-optimal solutions an approximation algorithm
- 17. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Table of Contents 2 The Approximate Algorithm 3 Applications 4 Some Questions
- 18. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Simulation
- 19. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Simulation
- 20. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Simulation
- 21. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Simulation
- 22. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Simulation
- 23. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Time Complexity O(V + E)
- 24. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation We say that an algorithm for a problem has an approximation ratio of ρ(n) if, for any input of size n, the cost X of the solution produced by the algorithm is within a factor of ρ(n) of the cost X∗ of an optimal solution: max( X X∗ , X∗ X )≤ ρ(n).
- 25. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(Is it a Vertex-Cover?)
- 26. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation
- 27. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation We need to prove that |X| ≤ 2|X∗ |
- 28. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation We need to prove that |X| ≤ 2|X∗ | We will first prove that |X∗ | ≥ |Y |
- 29. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(|X∗ | ≥ |Y |)
- 30. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(|X∗ | ≥ |Y |) : Another Subset of Arbitrary Edges
- 31. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(|X∗ | ≥ |Y |) : Another Subset of Arbitrary Edges
- 32. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(|X∗ | ≥ |Y |) : Another Subset of Arbitrary Edges
- 33. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(|X∗ | ≥ |Y |) : Another Subset of Arbitrary Edges
- 34. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation Now we will prove that, |X| = 2|Y |
- 35. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation(|X| = 2|Y |)
- 36. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Proof of 2-Approximation |X| = 2|Y | |Y | ≤ |X∗ | So,|X| ≤ 2|X∗ | thereby proving 2-Approximation
- 37. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Table of Contents 2 The Approximate Algorithm 3 Applications 4 Some Questions
- 38. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Applications
- 39. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Applications Selecting Optimal Position of Security Cameras.
- 40. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Applications Selecting Optimal Position of Security Cameras. Setting of ATM booths Optimally.
- 41. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Applications Selecting Optimal Position of Security Cameras. Setting of ATM booths Optimally. For Bipartitle Graph, Maximum Matiching = Minimum Vertex Cover.
- 42. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Table of Contents 2 The Approximate Algorithm 3 Applications 4 Some Questions
- 43. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Some Questions
- 44. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Some Questions Is there any graph where this approximate algorithm always gives twice the size of optimal solution?
- 45. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Some Questions 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?
- 46. Approximation Algorithm: The Vertex-cover Problem Sudipta Saha Shubha 1205014 Repon Kumar Roy 1205002 The Approximate Algorithm Applications Some Questions Thanks and Questions are Welcome