Generating Cut Conjunctions and Bridge Avoiding Extensions in Graphs

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.

This research was partially supported by the National Science Foundation (Grant IIS-0118635), and by DIMACS, the National Science Foundation’s Center for Discrete Mathematics and Theoretical Computer Science.