5.1 Integer Programming Formulation 5.2 Integrality Gap 5 Linear Programming Formulation and Integrality Gap

Abstract

\Multiway cuts in directed and node weighted graphs", 21st ICALP, 1994. LR88] T. Leighton and S. Rao, \An approximate max-ow min-cut theorem for uniform multi-commodity ow problems with applications to approximation algorithms," 29th FOCS, pp. 422{431, 1988. Directed graphs are dealt with in manuscript, Feb., 1992. 16 to the fes problem costs at most = 3 times the cost of an optimal solution to the corresponding fvs problem. By Claim 14, the cost of this fes problem is n=2 ? 1; this yields that the cost of the fvs problem is at least n=6 ? 1=3 = (n). 2 We show an (log k) integrality gap for the subset-fvs and subset-fes problems in the following graph F n , which is the union of two graphs: H k = (S; E) on k special vertices, as deened above, and a clique on the remaining n ? k vertices. To make F n connected we connect H k to the clique by a single edge. Note that no cycle that goes through a vertex in S intersects the clique. This fact, together with the integrality gap shown above, imply the (log k) integrality gap for the subset-fvs and subset-fes problems. Acknowledgement The bootstrapping technique was developed jointly with Madhu Sudan. We thank Madhu Sudan for his permission to include it here. We would like to thank Naveen Garg for useful discussions. \Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference", 5th SODA, pp. 344{353, 1994. BG94] A. Becker and D. Geiger, \Approximation algorithms for the loop cutset problem", 13th 15 fvs problem in undirected graph, we can nd a constant factor approximation in polynomial time BGNR94, BG94, BFB94]. Interestingly, Goemans and Williamson recently gave an alternative integer programming formulation for the fvs problem for which the gap is two, see H95]. Let G = (V; E) be an undirected graph. In this section we assume that all weights in G are unit weights. Let jV j = n, and let the set of special vertices be S = fs 1 ; : : :; s k g. We denote by C the set of cycles in G, and by C v the set of cycles passing through a particular vertex v. Let t : V ! 0; 1] be an indicator variable for membership in a feedback set of G. The following is a …

Cite this paper

@inproceedings{Bollob199651IP, title={5.1 Integer Programming Formulation 5.2 Integrality Gap 5 Linear Programming Formulation and Integrality Gap}, author={B . Bollob and Elias Dahlhaus and David S. Johnson and Christos H. Papadimitriou and Mihalis Yannakakis}, year={1996} }