NC ALGORITHMS FOR COMPUTING A PERFECT MATCHING AND A MAXIMUM FLOW IN ONE-CROSSING-MINOR-FREE GRAPHS

作     者：Eppstein, David Vazirani, Vijay V. 

作者机构：Univ Calif Irvine Comp Sci Dept Irvine CA 92697 USA 

出 版 物：《SIAM JOURNAL ON COMPUTING》 (工业与应用数学会计算杂志)

年 卷 期：2021年第50卷第3期

页      面：1014-1033页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：NSF [CCF1815901] 

主　　题：parallel algorithms perfect matching graph minors mimicking networks maximum flow 

摘      要：In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in K-3,K-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,K-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 K-3,K-3-minor-free graphs and K-5-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.