NC Algorithms for Computing a Perfect Matching, the Number of Perfect Matchings, and a Maximum Flow in One-Crossing-Minor-Free Graphs

Abstract

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in K 3,3 -minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K 3,3 -free graphs." In this paper, we finally settle this 30-year-old open problem. Building on the recent breakthrough result of Anari and Vazirani giving an NC algorithm for finding a perfect matching in planar graphs and graphs of bounded genus, we also obtain NC algorithms for any minor-closed graph family that forbids a one-crossing graph. The class contains several well-studied graph families including the K 3,3 -minor-free graphs and K 5 -minor-free graphs. Graphs in these classes not only have unbounded genus, but also can have genus as high as O(n). In particular, we obtain NC algorithms for:• Determining whether a one-crossing-minor-free graph has a perfect matching and if so, finding one.• Finding a minimum weight perfect matching in a one-crossing-minor-free graph, assuming that the edge weights are polynomially bounded.• Finding a maximum st-flow in a one-crossing-minor-free flow network, with arbitrary capacities.The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.The K 3,3 -minor-free graphs are particularly attractive as a target for this problem because they form a natural extreme case for certain approaches. In particular, they are known to have Pfaffian orientations, by which their matchings can be counted using matrix determinants [22,25], while for K 3,3 itself and for any minor-free family that does not forbid it, this tool is unavailable. However, our result breaks through this barrier: we give an NC algorithm for finding a perfect matching in graphs belonging to any one-crossing-minor-free class of graphs. That is, if H is any graph that can be drawn in the plane with only one crossing pair of edges, then we can find perfect matchings in the H-minor-free graphs in NC. Because K 3,3 can be drawn with one crossing, our result includes in particular the K 3,3 -minor-free graphs. More generally, we can find in NC a perfect matching of minimum weight, in the same families of graphs, when the weights are polynomially-bounded integers.In another direction, we obtain an NC algorithm for finding a maximum st-flow in any flow network whose underlying undirected graph belongs to a one-crossing-minor-free family. This generalizes Johnson's 1987 result [19], showing that maximum st-flow in a planar network is in NC. Technical ideasOur main new technical idea is that of a matching-mimicking network. Given a graph G and a set T of terminal vertices, a matching-mimicking network is a graph G , containing T , that has the same pattern of matchings: every matching of G that covers G \ T corresponds to a matching of G that covers G \ T and v...

Planar Graph Perfect Matching Is in NC

“…Remark 2. Very recent work [EV18] has resolved the open problem stated above about K 3,3 -free graphs.…”

Parameterizing the Permanent: Hardness for $K_8$-minor-free graphs

“…On the positive side, Curticapean [7] and Eppstein and Vazirani [13] lifted the algorithms for K 3.3 -minorfree and K 5 -minor-free graphs to H-minor-free graphs for any graph H that can be drawn in the plane with a single crossing, for example, H = K 3,3 and H = K 5 . (Note that excluding single-crossing minors H yields different graph classes than the class of single-crossing graphs themselves, for which an immediate reduction to the FKT method is possible.)…”

Domain Sparsification of Discrete Distributions using Entropic Independence

“…The implementation of each iteration of the Markov chain involves counting perfect matchings over S ⊆ V, and we do not have a poly(k) time algorithm for counting matchings in general graphs. We thus only consider downward closed graph families with an FPRAS for counting perfect matchings, e.g., bipartite graphs [JSV04], planar graphs [Kas67], certain minor-free graphs [EV19], and small genus graphs [GL99]. Our main results imply that as long as we estimate the probability of every vertex being part of a random k-matching, we can reduce the task of sampling k-matchings on an n vertex graph to graphs with only n 3/4 • poly(k) many vertices.…”