We consider the following NP-hard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constant-factor approximation algorithm, based on the primal-dual method. Moreover, we show that the integrality gap of the natural LP relaxation of the problem is ( n), where n denotes the number of vertices in the graph.
These algorithms also lead to a characterization of the integrality gap The integrality gap of an integer programming formulation is the worst-case ratio between the optimum value of the integer program and the optimum value of its linear relaxation. of two different integer programming formulations of the problem, as we now explain. Let @math denote the collection of all the cycles @math of @math . A natural integer programming formulation for the feedback vertex set problem is as follows: (Throughout, @math denotes the (non-negative) cost of vertex @math .) The algorithm of Bar- @cite_1 implies that the integrality gap of this integer program is @math . Later, Even, Naor, Schieber, and Zosin @cite_0 proved that its integrality gap is also @math .
@cite_0 @cite_1
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We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed. We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1 2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center. Recently, Becker and Geiger and Bafna, Berman and Fujito gave 2-approximation algorithms for the feedback vertex set problem in undirected graphs. We show how their algorithms can be explained in terms of the primal-dual method for approximation algorithms, which has been used to derive approximation algorithms for network design problems. In the process, we give a new integer programming formulation for the feedback vertex set problem whose integrality gap is at worst a factor of two; the well-known cycle formulation has an integrality gap of @Q(logn), as shown by Even, Naor, Schieber and Zosin. We also give a new 2-approximation algorithm for the problem which is a simplification of the algorithm.