Excluding Single-Crossing Matching Minors in Bipartite Graphs

Archontia C. Giannopoulou, Dimitrios M. Thilikos, Sebastian Wiederrecht

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

By a seminal result of Valiant, computing the permanent of (0, 1)-matrices is, in general, #P-hard. In 1913 Pólya asked for which (0, 1)-matrices A it is possible to change some signs such that the permanent of A equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding K3, 3 as a matching minor. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond K3, 3. Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the permanent of the corresponding (0, 1)-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing-minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and K3, 3 which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains #P-hard on bipartite graphs which exclude K5, 5 as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes. As another application of our structure theorem, we obtain a strict generalisation of the algorithm for the k-vertex disjoint directed paths problem on digraphs of bounded directed treewidth.

Original languageEnglish (US)
Title of host publication34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
PublisherAssociation for Computing Machinery
Pages2111-2121
Number of pages11
ISBN (Electronic)9781611977554
StatePublished - 2023
Event34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023 - Florence, Italy
Duration: Jan 22 2023Jan 25 2023

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume2023-January

Conference

Conference34th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2023
Country/TerritoryItaly
CityFlorence
Period1/22/231/25/23

ASJC Scopus subject areas

  • Software
  • General Mathematics

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