Generating cut conjunctions and bridge avoiding extensions in graphs

Authors:
L. Khachiyan;E. Boros;K. Borys;K. Elbassioni;V. Gurvich;K. Makino
Affiliations:
Department of Computer Science, Rutgers University, Piscataway, NJ;RUTCOR, Rutgers University, Piscataway, NJ;RUTCOR, Rutgers University, Piscataway, NJ;Max-Planck-Institut für Informatik, Saarbrücken, Germany;RUTCOR, Rutgers University, Piscataway, NJ;Division of Mathematical Science for Social Systems, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan
Venue:
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Year:
2005

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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 enumerate all minimal edge sets X⊆E such that every vertex pair (s,t) ∈ B is disconnected in $(V,E \smallsetminus X)$, generalizing well-known efficient algorithms for enumerating all minimal s-t cuts, for a given pair s,t ∈ V of vertices. We also present an incremental polynomial time algorithm for enumerating all minimal subsets X⊆E such that no (s,t) ∈ B is a bridge in (V,X ∪ B). These two enumeration problems are special cases of the more general cut conjunction problem in matroids: given a matroid M on ground set S=E ∪ B, enumerate all minimal subsets X⊆E such that no element b ∈ B is spanned by $E \smallsetminus X$. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V,E ∪ B), the enumeration of cut conjunctions for vectorial matroids turns out to be NP-hard.