APPROXIMATING MINIMUM FEEDBACK SETS AND MULTICUTS IN DIRECTED-GRAPHS

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FV S) problem, and the weighted feedback edge set (FES) problem. In the F VS (resp. FES) problem, one is given a directed graph with weights (ea ch of which is at least one) on the vertices (resp, edges), and is ask ed to find a subset of vertices (resp. edges) with minimum total weigh t that intersects every directed cycle in the graph. These problems ar e among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: SUBSET-FVS and SUBSE T-FES, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, de noted by X. This generalization is also NP-hard even when \X\ = 2. We present approximation algorithms for the SUBSET-FVS and SUBSET-FES pro blems. The first algorithm we present achieves an approximation factor of O(log(2)\X\). The second algorithm achieves an approximation facto r of O (min(log tau log log tau*, log n log log n)}, where tau* is th e value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut p roblem in a special type of directed networks which we call circular n etworks, and show that the SUBSET-FES and SUBSET-FVS problems are equi valent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1 + epsilon) approximation to the fractional optimal feedback vertex set. Computing the approxima te solution is much simpler and more efficient than general linear pro gramming methods. All of our algorithms use this approximate solution.