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Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

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Integer Programming and Combinatorial Optimization (IPCO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12707))

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Abstract

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb (2004) introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy–Binet formula.

This paper studies counting on linear matroid problems extending Webb’s work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities.

We also observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey (2009) and Cheung–Lau–Leung (2014) can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities.

The full version of this paper is available at https://arxiv.org/abs/1912.00620.

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Notes

  1. 1.

    An equivalent definition of Pfaffian orientation is as follows: an orientation of G is Pfaffian if every even-length cycle C such that \(G-V(C)\) has a perfect matching has an odd number of edges directed in either direction.

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Acknowledgments

The authors thank Satoru Iwata for his helpful comments, and Yusuke Kobayashi, Yutaro Yamaguchi, and Koyo Hayashi for discussions. This work was supported by JST ACT-I Grant Number JPMJPR18U9, Japan, and Grant-in-Aid for JSPS Research Fellow Grant Number JP18J22141, Japan.

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuki Matoya & Taihei Oki

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  1. Kazuki Matoya
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Correspondence to Taihei Oki .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Mohit Singh

  2. Cornell University, Ithaca, NY, USA

    David P. Williamson

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Matoya, K., Oki, T. (2021). Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems. In: Singh, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2021. Lecture Notes in Computer Science(), vol 12707. Springer, Cham. https://doi.org/10.1007/978-3-030-73879-2_16

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  • DOI: https://doi.org/10.1007/978-3-030-73879-2_16

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