2018/02/28 by David Eppstein, Vijay V. Vazirani
Computer Science · #3-dimensional matching #Advanced Graph Theory Research #Blossom algorithm #Complexity and Algorithms in Graphs #Computational Geometry and Mesh Generation #Graph #Matching (statistics) #Maximum flow problem #Pathwidth #Perfect graph #Perfect graph theorem #Planar graph #Trivially perfect graph #cs.DS
paper · pdf · doi:10.1137/19m1256221
published as SIAM J. Computing 50 (3): 1014-1033, 2021 · 21 pages, 6 figures
arxiv created 2020/05/05 · openalex created_date 2020/05/13 · openalex publication_date 2021/01/01 · arxiv updated 2021/06/29 · openalex updated_date 2026/08/06
In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in K3,3-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 K3,3-free graphs.” In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph. This family includes several well-studied graph families including the K3,3-minor-free graphs and K5-minor-free graphs. Graphs in these families not only have unbounded genus, but can have genus as high as O(n). Our method applies as well to several other problems related to perfect matching. In particular, we obtain NC algorithms for the following problems in any family of graphs (or networks) with a one-crossing forbidden minor: (1) Determining whether a given graph has a perfect matching and, if so, finding one. (2) Finding a minimum-weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. (3) Finding a maximum st-flow in the 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.