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

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)verticesundirected graphedgeNP-completeNIST

Vertex Cover Example Determine the smallest subset of vertex that “Cover” the graph on the right

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

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 > = CA in polynomial time. then this is a approximation solution

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)rmuhamma

Euclidean TSP Example TempP Approx T MST

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

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 GT’<= CT

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