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Counting matchings via capacity-preserving operators

Part of: Inequalities Graph theory Polynomials, rational functions Combinatorics

Published online by Cambridge University Press:  30 April 2021

Leonid Gurvits  and
Jonathan Leake
Leonid Gurvits
Affiliation:
City College of New York, New York, NY, USA
Jonathan Leake*
Affiliation:
KTH, Stockholm, Sweden
*
*Corresponding author. Email: jonathan@jleake.com
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Abstract

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The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

MSC classification

Primary: 05A20: Combinatorial inequalities
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 26C10: Polynomials 26D10: Inequalities involving derivatives and differential and integral operators
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 6 , November 2021 , pp. 956 - 981
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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