# Approximating minimum feedback sets and multi-cuts in directed graphs

## Abstract

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 (FES). 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 SUBSET-FVS, 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^{*} log log τ^{*}, log *n* log log *n*)}, where τ^{*} denotes the value of the optimum fractional solution of the problem at hand. For the SUBSET-FVS and SUBSET-FVS problems we also give an algorithm that achieves an approximation factor of *O*(log^{2} ‖*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 *combinatorial* 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.

## Keywords

Approximation Algorithm Approximation Factor Directed Cycle Fractional Solution Circular Network## Preview

Unable to display preview. Download preview PDF.

## References

- [1]M. Abramovici, M.A. Breuer and A.D. Friedman, “Digital Systems Testing and Testable Design,” New York, Computer Science Press, 1990.Google Scholar
- [2]N. Garg, V.V. Vazirani and M. Yannakakis, “Approximate max-flow min-(multi) cut theorems and their applications,”
*25th STOC*, pp. 698–707, 1993.Google Scholar - [3]R. Gupta, R. Gupta and M.A. Breuer, “BALLAST: A Methodology for Partial Scan Design,”
*Proc. 19th Int'l. Symp. on Fault-Tolerant Computing*,' pp. 118–125, June, 1989.Google Scholar - [4]T.C. Hu, “Multi-commodity network flows,”
*Operations Research*, 11, pp. 344–360, 1963.Google Scholar - [5]R.M. Karp, “Reducibility among combinatorial problems,”
*Complexity of Computer Computations*, pp. 85–104, Plenum Press, N.Y., 1972.Google Scholar - [6]P.N. Klein, S.A. Plotkin, S. Rap and É. Tardos, “New network decompositions theorems with applications,” unpublished manuscript, 1993.Google Scholar
- [7]P. Klein, A. Agrawal, R. Ravi, and S. Rao, “Approximation through multi-commodity flow,”
*31st FOCS*, pp. 726–737, 1990.Google Scholar - [8]P. Klein, C. Stein, and É. Tardos, “Leighton-Rao might be practical: faster approximation algorithms for concurrent flow with uniform capacities,”
*22nd STOC*, pp. 310–321, 1990.Google Scholar - [9]A. Kunzmann and H.J. Wunderlich, “An Analytical Approach to the Partial Scan Problem,”
*Journal of Elec. Testing: Theory and Applications*, 1, pp. 163–174, 1990.Google Scholar - [10]M. Luby and N. Nisan, “A parallel approximation algorithm for positive linear programming,”
*25th STOC*, pp. 448-457, 1993.Google Scholar - [11]T. Leighton and S. Rao, “An approximate max-flow min-cut theorem for uniform multi-commodity flow problems with applications to approximation algorithms,”
*29th FOCS*, pp. 422–431, 1988. Directed graphs are dealt with in manuscript, Feb., 1992.Google Scholar - [12]C.E. Leiserson and J.B. Saxe, “Retiming Synchronous Circuitry,”
*Algorithmica*, Vol. 6, No. 1, pp. 5–35. 1991.Google Scholar - [13]S. Plotkin, É. Tardos and D. Shmoys, “Fast approximation algorithms for fractional packing and covering problems”,
*32nd FOCS*, pp. 495–504, 1991.Google Scholar - [14]P.D. Seymour, “Packing Directed Circuits Fractionally,” Manuscript, (1992). To appear in
*Combinatorica*.Google Scholar