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 $K_{3,3}$ 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 $K_{3,3}$ 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 $K_{5,5}$ 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.
This is joint work with Archontia Giannopoulou and Dimitrios Thilikos.
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