Enumerating Cut Conjunctions in Graphs and Related Problems

LetG = (V,E) be an undirected graph, and letB ⊆ 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,ErX), 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 rX. 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. Acknowledgements: 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.