Approximating minimum feedback sets andmulti-cuts in directed graphs

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set problem (FBS). In the FVS (resp. FES) problem, one is given a directed graph with weights on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-Hard problems and have many applications. We also consider a generalization of these problems: SUBSET-FVS and SUBSBT-FZS, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset contains all the cycles that go through a distinguished input subset of vertices and edges. We present approximation algorithms for all four problems that achieve an approximation factor of O(min{log τ loglog τ *,log nloglog n)}, where τ * denotes the value of the optimum fractional solution of the problem at hand. For the SUBSBT-FVS and SUBSET-FBS problems we also give an algorithm that achieves an approximation factor of O(log² |X|), where X is the subset of distinguished vertices and edges. This algorithm is based on an approximation algorithm for the multi-cut problem in a special type of directed networks. Another contribution of our paper is a combznatomal algorithm that computes a (1 + ε) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.