This paper deals with approximating feedbacksets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set problem (FES). In the FVS (resp. FES) problem, one is given a directed graph with weights (each of which at least 1) 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 SUBSET-FES, 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, denoted by X. This generalization is also NP-Hard even when |X| = 2. We present approximation algorithms for the SUBSET-FVS and SUBSET-FES problems. The first algorithm we present achieves an approximation factor of O(log2|X|). The second algorithm achieves and approximation factor of O(min{log tau* loglog tau*, log n loglog n)}, where tau* is the value of teh optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the SUBSET-FES and SUBSET-FVS problems are equivalent 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.

By:* G. Even (Technion, Israel), J. Naor (Technion, Israel), B. Schieber and M. Sudan *

Published in: RC20074 in 1995

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