Abstract
In this paper, we define and study a natural generalization of the multicut and multiway cut problems: the minimum multi-multiway cut problem. The input to the problem is a weighted undirected graph G=(V,E) and k sets S 1,S 2,...,S k of vertices. The goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets S 1,S 2,...,S k , i.e., disconnects every pair of vertices u and v such that u,v∈ S i , for some i. This problem generalizes both the multicut problem, when |S i |=2, for 1≤ i≤ k, and the multiway cut problem, when k=1.
We present an approximation algorithm for the multi-multiway cut problem with an approximation ratio which matches that obtained by Garg, Vazirani, and Yannakakis [GVY96] on the standard multicut problem. Namely, our algorithm has an O(log 2k) approximation ratio. Moreover, we consider instances of the minimum multi-multiway cut problem which are known to have an optimal solution of light weight. We show that our algorithm has an approximation ratio substantially better than O(log 2k) when restricted to such “light” instances. Specifically, we obtain an O(log LP)-approximation algorithm for the problem, when all edge weights are at least 1, where LP is the value of a natural LP-relaxation of the problem. The latter improves the O(log LP loglog LP) approximation ratio for the minimum multicut problem (implied by the work of Seymour [Sey95] and Even et al. [ENSS98].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Chrikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (2003)
Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for multiway cut. In: Proceedings of the 30th Ann. ACM Symp. on Theory of Comp., ACM-SIAM (1998)
Cheriyan, J., Karloff, H.J., Rabani, Y.: Approximating directed multicuts. In: Proceedings of IEEE Symposium on Foundations of Computer Science (2001)
Cunningham, W.H., Tang, L.: Optimal 3-terminal cuts and linear programming. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 114–125. Springer, Heidelberg (1999)
Demaine, E.D., Immorlica, N.: Correlation clustering with partial information. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 1–13. Springer, Heidelberg (2003)
Dahlhaus, R., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM Journal on Computing 23, 864–894 (1994)
Downey, R.: Parameterized complexity for the skeptic. In: Proceedings of 18th IEEE Conference on Computational Complexity (2003)
Even, G., Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)
Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Canadian Journal of Mathematics 8, 399–404 (1956)
Feige, U., Goemans, M.X.: Approximating the value of two prover proof systems with applications to Max-2-Sat and Max-Dicut. In: Proceedings of the 3rd Israel Symposium on Theory of Computing and Systems, pp. 182–189 (1995)
Gupta, A.: Improved results for directed multicut. In: Proceedings 14th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 454–455. ACM-SIAM (2003)
Garg, N., Vazirani, V., Yannakakis, M.: Approximate max-flow min- (multi)cut theorems and their applications. SIAM Journal on Computing 25(2), 235–251 (1996)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of ACM 42, 1115–1145 (1995)
Karger, D.R., Klein, P., Stein, C., Thorup, M., Young, N.E.: Rounding algorithms for a geometric embedding of minimum multiway cut. In: Proceedings of ACM Symposium on the Theory of Computing, pp. 668–678 (1999)
Klein, P.N., Plotkin, S.A., Rao, S., Tardos, S.E.: New network decompositions theorems with applications. Brown University Technical Report CS-93-30 (1993)
Klein, P., Rao, S., Agrawal, A., Ravi, R.: An approximate max-flow min-cut relation for undirected multicommodity flow, with applications. Combinatorica 15, 187–202 (1995)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8, 261–277 (1988)
Leighton, F.T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46(6), 787–832 (1999)
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms 31, 355–354 (1999)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)
Seymour, P.D.: Packing directed circuits fractionally. Combinatorica 15(2), 281–288 (1995)
Shmoys, D.: Cut problems and their applications to divide-and-conquer. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-Hard Problems, pp. 192–235. PWS Publishing (1996)
Yannakakis, M., Kanellakis, P.C., Cosmadakis, S.C., Papadimitriou, C.H.: Cutting and partitioning a graph after a fixed pattern. In: Díaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 712–722. Springer, Heidelberg (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Avidor, A., Langberg, M. (2004). The Multi-multiway Cut Problem. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-27810-8_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22339-9
Online ISBN: 978-3-540-27810-8
eBook Packages: Springer Book Archive