An efficient network flow code for finding all minimum cost s–t cutsets

Lecture Notes in Computer Science

Gurvich

et al. 2004

Abstract. Let G = (V, E) be a (directed) graph with vertex set V and edge (arc) set E. Given a set P 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 P. For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. For directed graphs both of these problems are NP-hard. We also give a polynomial delay algorithm for enumerating minimal sets of arcs connecting respectively two given nodes s1 and s2 to a given vertex t1, and each vertex of a given subset of vertices T2.

Section: Cut-disjunction (Cd): S I Is Not Connected To T I In the (Dimentioning

confidence: 99%

Generating Paths and Cuts in Multi-pole (Di)graphs

Lecture Notes in Computer Science

Gurvich

et al. 2004

“…For instance, it is known [19] that the problems of listing all minimal cuts or all spanning trees of an undirected graph G = (V, E) can be solved with delay O(|E|) per generated cut or spanning tree. It is also known (see e.g., [7,10,18]) that all minimal (s, t)-cuts or (s, t)-paths, can be listed with delay O(|E|) per cut or path. Furthermore, polynomial delay algorithms also exist for listing perfect matchings, maximal matchings, maximum matchings in bipartite graphs, and maximal matchings in general graphs, see e.g.…”

Section: Enumeration Algorithmsmentioning

confidence: 99%

Algorithms for Generating Minimal Blockers of Perfect Matchings in Bipartite Graphs and Related Problems

Lecture Notes in Computer Science

Gurvich

2004

Abstract. A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, this gives a polynomial delay algorithm for listing the anti-vertices of the perfect matching polytope P (G) = {x ∈ R E | Hx = e, x ≥ 0}, where H is the incidence matrix of G. We also give similar generation algorithms for other related problems, including d-factors in bipartite graphs, and perfect 2-matchings in general graphs.

“…For instance, it is known [18] that the problems of listing all minimal cuts or all spanning trees of an undirected graph G = (V , E) can be solved with delay O(|E|) per generated cut or spanning tree. It is also known (see e.g., [8,12,17]) that all minimal (s, t)-cuts or (s, t)-paths, can be listed with delay O(|E|) per cut or path, both in the directed and undirected cases. Furthermore, if π(X) is the property that the subgraph (V , X) of a directed graph G = (V , E) contains a directed cycle, then F π is the family of minimal directed circuits of G, while F d π consists of all minimal feedback arc sets of G (i.e.…”

Section: Introductionmentioning

confidence: 99%

On Enumerating Minimal Dicuts and Strongly Connected Subgraphs

Khachiyan¹,

et al. 2007

Algorithmica

We consider the problems of enumerating all minimal strongly connected subgraphs and all minimal dicuts of a given strongly connected directed graph G = (V , E). We show that the first of these problems can be solved in incremental polynomial time, while the second problem is NP-hard: given a collection of minimal dicuts for G, it is NP-hard to tell whether it can be extended. The latter result implies, in particular, that for a given set of points A ⊆ R n , it is NP-hard to generate all maximal subsets of A contained in a closed half-space through the origin. We also discuss the enumeration of all minimal subsets of A whose convex hull contains the origin as an interior point, and show that this problem includes as a special case the well-known hypergraph transversal problem.