Counting Matchings via Capacity Preserving Operators

TL;DR

This paper develops a unified theory of capacity preserving operators for polynomials, providing tight bounds and applying it to prove a recent result on lower bounds for matchings, advancing combinatorial and algebraic methods.

Contribution

It introduces a comprehensive framework for capacity preservation in real stability preservers, leading to new proofs of combinatorial bounds.

Findings

Established tight bounds for capacity preservation by all nondegenerate real stability preservers.

Provided a new proof of Friedland's lower matching conjecture using capacity preservation theory.

Unified and extended previous results on capacity bounds and combinatorial quantities.

Abstract

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 utilized 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 Csikv\'ari, which settled Friedland's lower matching conjecture.