A Generalization of Permanent Inequalities and Applications in Counting and Optimization

A polynomial $p\in\mathbb{R}[z1,\dots,zn]$ is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the $z1z2\dots z_n$ monomial of a real stable polynomial $p$ with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems. Here, we study a more general question: Given a stable multilinear polynomial $p$ with nonnegative coefficients and a set of monomials $S$, we show that if the polynomial obtained by summing up all monomials in $S$ is real stable, then we can lowerbound the sum of coefficients of monomials of $p$ that are in $S$. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijver's inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh, and give deterministic polynomial time approximation algorithms for several counting problems.