Abstract
We show that enumerating all minimal spanning and connected subsets of a given matroid can be solved in incremental quasi-polynomial time. In the special case of graphical matroids, we improve this complexity bound by showing that all minimal 2-vertex connected edge subsets of a given graph can be generated in incremental polynomial time.
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.
References
Tamura, A., Shioura, A., Uno, T.: An optimal algorithm for scanning all spanning trees of undirected graphs. SIAM Journal on Computing 26(3), 678–692 (1997)
Bang-Jensen, J., Gabow, H.N., Jordán, T., Szigeti, Z.: Edge-connectivity augmentation with partition constraints. SIAM Journal on Discrete Mathematics 12(2), 160–207 (1999)
Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: An inequality for polymatroid functions and its applications. Discrete Applied Mathematics 131, 255–281 (2003)
Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L.: Enumerating minimal dicuts and strongly connected subgraphs and related geometric problems. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 152–162. Springer, Heidelberg (2004)
Coulbourn, C.J.: The Combinatorics of Network Reliability. Oxford University Press, Oxford (1987)
Eiter, T., Gottlob, G.: Identifying the minimal transversals of a hypergraph and related problems. SIAM Journal on Computing 24, 1278–1304 (1995)
Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. Journal of Algorithms 21, 618–628 (1996)
Ishii, T., Nagamochi, H., Ibaraki, T.: Optimal augmentation to make a graph k-edge-connected and triconnected. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, United States, pp. 280–289 (1998)
Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Makino, K.: Generating cut conjunctions and bridge avoiding extensions in graphs. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 156–165. Springer, Heidelberg (2005)
Lawler, E., Lenstra, J.K., Kan, A.H.G.R.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM Journal on Computing 9, 558–565 (1980)
Lovasz, L.: Submodular functions and convexity. In: Grotschel, M., Bachem, A., Korte, B. (eds.) Mathematical Programming: The State of the Art, New York, pp. 235–257. Springer, Heidelberg (1983)
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)
Schwikowski, B., Speckenmeyer, E.: On enumerating all minimal solutions of feedback problems. Discrete Applied Mathematics 117, 253–265 (2002)
Valiant, L.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8, 410–421 (1979)
Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Makino, K. (2006). Enumerating Spanning and Connected Subsets in Graphs and Matroids. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_41
Download citation
DOI: https://doi.org/10.1007/11841036_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-38875-3
Online ISBN: 978-3-540-38876-0
eBook Packages: Computer ScienceComputer Science (R0)