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.
A better formulation has been introduced by @cite_5 . For @math , denote by @math the set of the edges of @math having both ends in @math , by @math the subgraph of @math induced by @math , and by @math the degree of @math in @math . Then, the following is a formulation for the feedback vertex set problem: @cite_5 showed that the integrality gap of this integer program asymptotically equals @math . Constraints ) derive from the simple observation that the removal of a feedback vertex set @math from @math generates a forest having at most @math edges. Notice that the covering inequalities ) are implied by ).
@cite_5
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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. 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. A problem arising in the design of vacuum systems and having applications to some natural problems of interconnection design is described as follows. (1) Given a set X and subsets @math of @math , satisfying @math , find a graph G with vertex set X and the minimum number of edges such that for any i, the subgraph induced by @math has a connected component containing @math .Two other problems related to this one are the following ones. (2) Given a set X and subsets @math such that @math , find a graph G with vertex set X and the minimum number of edges such that for any i the subgraph @math induced by @math in G is connected. (3) Given a set X and subsets @math such that @math , find a graph G with vertex set X, find a graph G with vertex set X and the minimum number of edges such that for any subset I of @math , the subgraph induced by @math is co... 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.