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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Let G=(V,E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let @math be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimum weight. This problem has a variety of applications, among them genetic linkage analysis and circuit testing. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NP-complete and also generalizes the multiway cut problem. We provide a polynomial time algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. This implies a @math -approximation algorithm for the subset feedback vertex set problem, where @math is the maximum degree in G. We also consider the multicut problem and show how to achieve an @math approximation factor for this problem, where @math is the value of the optimal fractional solution. To achieve the @math factor we employ a bootstrapping technique. A feedback vertex set of an undirected graph is a subset of vertices that intersects with the vertex set of each cycle in the graph. Given an undirected graph G with n vertices and weights on its vertices, polynomial-time algorithms are provided for approximating the problem of finding a feedback vertex set of G with smallest weight. When the weights of all vertices in G are equal, the performance ratio attained by these algorithms is 4-(2 n). This improves a previous algorithm which achieved an approximation factor of @math for this case. For general vertex weights, the performance ratio becomes @math where @math denotes the maximum degree in G. For the special case of planar graphs this ratio is reduced to 10. An interesting special case of weighted graphs where a performance ratio of 4-(2 n) is achieved is the one where a prescribed subset of the vertices, so-called blackout vertices, is not allowed to participate in any feedback vertex set. It is shown how these algorithms can improve the search performance for constraint satisfaction problems. An application in the area of Bayesian inference of graphs with blackout vertices is also presented.