Abstract
Let G=(V,E) be an undirected graph, and let B⊆V×V be a collection of vertex pairs. We give an incremental polynomial time algorithm to generate all minimal edge sets X⊆E such that every pair (s,t)∈B of vertices is disconnected in (V,E ∖ X), generalizing well-known efficient algorithms for generating all minimal s-t cuts, for a given pair s,t of vertices. We also present an incremental polynomial time algorithm for generating all minimal subsets X⊆E such that no (s,t)∈B is a bridge in (V,X∪B). Both above problems are special cases of a more general problem that we call generating cut conjunctions for matroids: given a matroid M on ground set S=E∪B, generate all minimal subsets X⊆E such that no element b∈B is spanned by E ∖ X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V,E∪B), the more general problem of generating cut conjunctions for vectorial matroids turns out to be NP-hard.
Article PDF
Similar content being viewed by others
References
Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K.: Generating paths and cuts in multi-pole (di)graphs. In: Fiala, J., Koubek, V., Kratochvil, J. (eds.) Mathematical Foundations of Computer Science MFCS, Prague, Czech Republic, August 22–27, 2004. Lecture Notes in Computer Science, vol. 3153, pp. 298–309. Springer, Berlin (2004)
Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of some enumeration problems for matroids. SIAM J. Discrete Math. 19(4), 966–984 (2005)
Boros, E., Elbassioni, K., Gurvich, V.: Transversal hypergraphs to perfect matchings in bipartite graphs: characterization and generation algorithms. J. Graph Theory 53(3), 209–232 (2006)
Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiway cuts. In: Proceedings of the 24th ACM Symposium on Theory of Computing, pp. 241–251 (1992)
Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM J. Comput. 24, 1278–1304 (1995)
Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21, 618–628 (1996)
Hu, T.C.: Multicomodity network flows. Oper. Res. 11, 344–360 (1963)
Johnson, D.S., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27, 119–123 (1988)
Lawler, E., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9, 558–565 (1980)
Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992)
Read, R.C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5, 237–252 (1975)
Schrijver, A.: Combinatorial Optimization Polyhedra and Efficiency, vol. B. Springer, Berlin (2003). p. 654
Schwikowski, B., Speckenmeyer, E.: On enumerating all minimal solutions of feedback problems. Discrete Appl. Math. 117(1–3), 253–265 (2002)
Shioura, A., Tamura, A.: Efficiently scanning all spanning trees of an undirected graph. J. Oper. Res. 38, 331–344 (1995)
Tarjan, R.: A note on finding the bridges of a graph. Inf. Process. Lett. 2, 160–161 (1974)
Tsukiyama, S., Shirakawa, I., Ozaki, H., Ariyoshi, H.: An algorithm to enumerate all cutsets of a graph in linear time per cutset. J. Assoc. Comput. Mach. 27, 619–632 (1980)
Vazirani, V.: Approximation Algorithms. Springer, Berlin (2001)
Welsh, D.J.A.: Matroid Theory. Academic, London (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the National Science Foundation (Grant IIS-0118635), by DIMACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical Computer Science, and by the Scientific Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan.
Our friend and colleague, Leonid Khachiyan passed away with tragic suddenness while we were preparing this manuscript.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Khachiyan, L., Boros, E., Borys, K. et al. Generating Cut Conjunctions in Graphs and Related Problems. Algorithmica 51, 239–263 (2008). https://doi.org/10.1007/s00453-007-9111-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9111-9