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WK15. Vertex Cover and Approximation Algorithm

WK15. Vertex Cover and Approximation Algorithm. By Lin, Jr-Shiun Choi, Jae Sung. Vertex Cover. Definition

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WK15. Vertex Cover and Approximation Algorithm

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  1. WK15. Vertex Cover and Approximation Algorithm By Lin, Jr-Shiun Choi, Jae Sung

  2. Vertex Cover • Definition A set of vertices in an undirected graph where every edge connects at least one vertex. The vertex cover problem is to find a minimum size set and is NP-complete. (NIST)

  3. Vertex Cover 2 • Example Determine the smallest subset of vertex that “Cover” the graph on the right. 1 3 4 5

  4. Vertex Cover 2 • Example Determine the smallest subset of vertex that “Cover” the graph on the right. ANS: { 1,3, 4 } 1 3 4 5

  5. Approximation Algorithm • Minimization problem if F , C > 1 such that you can find a solution which is <= CA in polynomial time. then this is a approximation solution • Maximization problem if F , C >< 1 such that you can find a solution which is >= CA in polynomial time. then this is a approximation solution

  6. Approximation Algorithm for Vertex Cover • Algorithm 1. choose a edge A, kick out all edges which connect to 2 ends of A ( include A). 2. choose other edges and repeat step1 until all edges are kick out. vertex-cover that is atmost twice the size of an optimal cover (rmuhamma)

  7. Euclidean TSP Example TempP Approx T MST

  8. Euclidean TSP Let Optimal TSP = T Let MST = M (TSP=spanning tree that visits all vertex) M should be smaller than T T >=M TempP=2m Approx T <=2M<=2T Approx T <= (1+ X )T X is any small number, e.g. 0.000001

  9. Approximation Algorithm for TSP • TSP is general graph that can not be approximated. Claim: an algorithm can solve it. we will show that approximation algorithm can be used to solve the Hamiltonian cycle problem G T’<= CT

  10. Approximation Algorithm for TSP CN+1 Add weight G Approximation Algorithm CN+1 <= CT 5N+1<=5T (N= # of Vertex) 26<=5T

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