# Approximation algorithms for feasible cut and multicut problems

## Abstract

Let *G=(V, E)* be an undirected graph with a capacity function *u:E→ℜ*_{+} and let *S*_{1}, *S*_{2},⋯, *S*_{ k } be *k* commodities, where each S_{i} consists of a pair of nodes. A set *S* of nodes is called feasible if it contains no *S*_{ i }, and a cut (*S, ¯S*) is called feasible if *S* is feasible. We show that several optimization problems on feasible cuts are NP-hard. We give a (4 ln 2)-approximation algorithm for the minimum capacity feasible v^{*}-cut problem. The multicut problem is to find a set of edges *F*\(\subseteq\)*E* of minimum capacity such that no connected component of *G/F* contains a commodity *S*_{ i }. We show that an *α*-approximation algorithm for the minimum-ratio feasible cut problem gives a 2*α(1+ln T*)-approximation algorithm for the multicut problem, where *T* denotes the cardinality of \(\cup _i S_i\). We give a new approximation guarantee of *O(t* log *T*) for the minimum capacity-to-demand ratio Steiner cut problem; here each *S*_{ i } is a set of nodes and *t* denotes the maximum cardinality of a commodity *S*_{ i }.

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## References

- [AD 91]D. Avis and M. Deza,
*The cut cone, L*_{1}*embeddability, complexity, and multicommodity flows*, Networks, 21 (1991), pp. 595–617.Google Scholar - [BV 94]D. Bertsimas and R. Vohra,
*Linear programming relaxations, approximation algorithms and randomization: a unified view of covering problems*, manuscript, 1994.Google Scholar - [BTV 95]D. Bertsimas, C. Teo and R. Vohra,
*Nonlinear formulations and improved randomized approximation algorithms for multicut problems*, Proc. 4th I.P.C.O, pp. 29–39, LNCS 920, Springer-Verlag, Berlin, 1995.Google Scholar - [Ch 79]V. Chvátal,
*A greedy heuristic for the set-covering problem*, Mathematics of Operations Research, 4 (1979), pp. 233–235.Google Scholar - [DJPSY 94]E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour and M. Yannakakis,
*The complexity of multiterminal cuts*, SIAM J. Computing 23 (1994), pp. 864–894.Google Scholar - [GVY 93]N. Garg, V. Vazirani and M. Yannakakis,
*Approximate max-flow min-(multi)cut theorems and their applications*, Proc. 25th ACM S.T.O.C., 1993, pp. 698–707.Google Scholar - [GVY 94]N. Garg, V. Vazirani and M. Yannakakis,
*Multiway cuts in directed and node weighted graphs*, Proc. 21st I.C.A.L.P., LNCS 820, Springer-Verlag, Berlin, 1994.Google Scholar - [GM 84]M. Gondran and M. Minoux,
*Graphs and algorithms*, Wiley, New York, 1984.Google Scholar - [GLS 88]M. Grötschel, L. Lovász and A. Schrijver,
*Geometric algorithms and combinatorial optimization*, Springer-Verlag, Berlin, 1988.Google Scholar - [HFKI 87]S. Hashizume, M. Fukushima, N. Katoh and T. Ibaraki,
*Approximation algorithms for combinatorial fractional programming problems*, Mathematical Programming 37 (1987), pp. 255–267.Google Scholar - [HRW 92]F. K. Hwang, D. S. Richards and P. Winter,
*The Steiner tree problem*, North-Holland, Amsterdam, 1992.Google Scholar - [KRAR 90]P. Klein, S. Rao, A. Agrawal, and R. Ravi, Preliminary version in
*Approximation through multicommodity flow*, Proc. 31st IEEE F.O.C.S., 1990, pp. 726–737.Google Scholar - [KPRT 94]P. Klein, S. Plotkin, S. Rao, and E. Tardos,
*Approximation algorithms for Steiner and directed multicuts*, manuscript, 1994.Google Scholar - [LR 88]F. T. Leighton and S. Rao,
*An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms*, manuscript, September 1993. Preliminary version in Proc. 29th IEEE F.O.C.S., 1988, pp. 422–431.Google Scholar - [LLR 94]N. Linial, E. London and Y. Rabinovich,
*The geometry of graphs and some of its algorithmic applications*, manuscript, April 1995. Preliminary version in Proc. 35th IEEE F.O.C.S., 1994, pp. 577–591.Google Scholar - [RV 93]S. Rajagopalan and V. Vazirani,
*Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs*, Proc. 34th IEEE F.O.C.S., 1993, pp. 322–331.Google Scholar