Approximating Minimum Feedback Sets and Multicuts in Directed Graphs
 G. Even,
 J. (Seffi) Naor,
 B. Schieber,
 M. Sudan
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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 (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) 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 NPhard problems and have many applications. We also consider a generalization of these problems: subsetfvs and subsetfes, 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, denoted by X . This generalization is also NPhard even when X=2 . We present approximation algorithms for the subsetfvs and subsetfes problems. The first algorithm we present achieves an approximation factor of O(log ^{ 2 } X) . The second algorithm achieves an approximation factor of O(min{log τ ^{ * } log log τ ^{ * } , log n log log n)} , where τ ^{ * } is the 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 problem in a special type of directed networks which we call circular networks, and show that the subsetfes and subsetfvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) 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.
 Title
 Approximating Minimum Feedback Sets and Multicuts in Directed Graphs
 Journal

Algorithmica
Volume 20, Issue 2 , pp 151174
 Cover Date
 19980201
 DOI
 10.1007/PL00009191
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Keywords

 Key words. Feedback vertex set, Feedback edge set, Multicuts.
 Industry Sectors
 Authors

 G. Even ^{(A1)}
 J. (Seffi) Naor ^{(A2)}
 B. Schieber ^{(A3)}
 M. Sudan ^{(A4)}
 Author Affiliations

 A1. Department of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel. guy@eng.tau.ac.il., IL
 A2. Computer Science Department, Technion, Haifa 32000, Israel. naor@cs.technion.ac.il., IL
 A3. IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA. sbar@watson.ibm.com., US
 A4. IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA. madhu@watson.ibm.com., US