A Generalized Matrix-Tree Theorem for Pfaffian Pairs
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The matrix-tree theorem counts the number of spanning trees of undirected graphs as well as the number of column bases of totally unimodular matrices. Extending this theorem, Webb (2004) introduced the notion of Pfaffian pairs as a pair of totally unimodular matrices for which counting of their common bases is tractable. This paper consolidates a list of combinatorial structures that can be represented as Pfaffian pairs. This includes spanning trees, Euler tours in four-regular digraphs, arborescences, and perfect matchings in K_3,3-free bipartite graphs. We also present a strongly polynomial-time algorithm to count maximum weight common bases in weighted Pfaffian pairs. Our algorithm makes use of Frank's weight splitting lemma on the weighted matroid intersection. We further consider the counting problem for Pfaffian pairs under group-label constraints.
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