Generating Cut Conjunctions in Graphs and Related Problems

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 generate all minimal edge sets X ⊆ E such that every pair (s, t) ∈ B of vertices is disconnected in (V , E X), generalizing well-known efficient algorithms for generating all minimal s-t cuts, for a given pair s, t of vertices. We also present an incremental polynomial time algorithm for generating all minimal subsets X ⊆ E such that no (s, t) ∈ B is This research was partially supported by the National Science Foundation (Grant IIS-0118635), by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science, and by the Scientific Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan. Our friend and colleague, Leonid Khachiyan passed away with tragic suddenness while we were preparing this manuscript. (V , X ∪ B). Both above problems are special cases of a more general problem that we call generating cut conjunctions for matroids: given a matroid M on ground set S = E ∪ B, generate all minimal subsets X ⊆ E such that no element b ∈ B is spanned by E X. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V , E ∪ B), the more general problem of generating cut conjunctions for vectorial matroids turns out to be NP-hard.

Polynomial-Delay Enumeration of Large Maximal Matchings

“…The idea of our algorithms is as follows. Our algorithms are based on the supergraph technique, which is frequently used in designing efficient enumeration algorithms [4,8,15,16,19,25]. In this technique, we define a directed graph on the set of all solutions and the enumeration algorithm simply traverses this directed graph from an arbitrary solution.…”

Tight Lower Bounds for the Number of Inclusion-Minimal st-Cuts

“…Enumerating the cutsets between all pairs of nodes reduces to the problem of solving a system of linear equations [19]. The notion of cutset has been generalized to cut conjunctions in [17].…”

Enumerating minimal dominating sets in chordal bipartite graphs

“…In particular, if an enumeration algorithm runs in time that is polynomial in the size of the input graph plus the number of enumerated objects, then it is called output polynomial. A large amount of results have been dedicated to output polynomial enumeration algorithms, e.g., [1,2,[10][11][12]18,[24][25][26]28,[32][33][34], and for various enumeration problems it has been shown that no output polynomial time algorithm can exist unless P = NP [24,26,28].…”

“…Based on the supergraph technique for enumerating vertex subsets in graphs [2,23,25,32,34], Golovach et al [15] presented a flipping method to generate the out-neighbors of a node of the supergraph, i.e., to generate new minimal dominating sets from a parent dominating set. Using this flipping method they were able to give output polynomial algorithms for enumerating the minimal dominating sets of line graphs and graphs of large girth.…”