Enumerating disjunctions and conjunctions of paths and cuts in reliability theory

Let G=(V,E)G=(V,E) be a (directed) graph with vertex set VV and edge (arc) set E  . Given a set PP of source–sink pairs of vertices of G  , an important problem that arises in the computation of network reliability is the enumeration of minimal subsets of edges (arcs) that connect/disconnect all/at least one of the given source–sink pairs of PP. For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. Furthermore, under the assumption that PP consists of all pairs within a given vertex set, we also give incremental polynomial time algorithm for enumerating all minimal path disjunctions and cut conjunctions. For directed graphs, the enumeration problem for cut disjunction is known to be NP-complete. We extend this result to path conjunctions and path disjunctions, leaving open the complexity of the enumeration of cut conjunctions. Finally, we give a polynomial delay algorithm for enumerating all minimal sets of arcs connecting two given nodes s1s1 and s2s2 to, respectively, a given vertex t1t1, and each vertex of a given subset of vertices T2T2.