Counting Minimum (s,t)-Cuts in Weighted Planar Graphs in Polynomial Time

Abstract

Abstract. We give an O(nd + n log n) algorithm computing the number of minimum (s, t)-cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest s-t path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomial-time algorithm for unweighted graphs with both s and t lying on the outer face. Our results hold for all locations of s and t and weighted graphs, and have direct applications in image segmentation and other computer vision problems.

Counting and sampling minimum cuts in genus g graphs

“…Let Σ = Σ \ (s * ∪ t * ). Simply knowing that G is a DAG immediately gives us the following lemma which generalizes Claim 1 of [2]. While the original claim deals with single cycles in planar graphs, our lemma describes more general circulations.…”

Counting and sampling minimum cuts in genus g graphs

“…Our algorithm requires only a few simple assumptions on the input graph; every edge has positive capacity, and there exists a directed path from s to every vertex in G and a directed path from every vertex in G to t. We make the second assumption so that vertices with no effect on the connectivity of s and t do not influence the number of minimum (s, t)-cuts. See [2,Section 4].…”

“…As an alternative to the above procedure, our algorithm can perform the original contraction procedure of Bezáková and Friedlander [2] and then compute a new embedding ofG without loops onto a new surface of genus at most g [39], but we must consider loops anyway due to technicalities introduced by the procedure in Section 6. Note the minimum embedding of a connected graph is known to be cellular [59].…”

Contiguous Minimum Single-Source-Multi-Sink Cuts in Weighted Planar Graphs

“…In essence, the tour is cut into path segments by the tree A. [2] for the details of this step), obtaining a non-crossing tour and the corresponding contiguous minimum (s, T )-cut.…”

“…This implies that the problem is #P -complete for general graphs [13]. Recently, building on [1], a polynomial-time algorithm was developed for the single-source-single-sink variant for planar graphs [2], using, as the first step, the same reduction to the maximal antichains. However, the reduction can not be applied in the contiguous multi-sink case, as the contractions can "bypass" vertices lying in the region defined by the contracted component.…”