A Generalization of Permanent Inequalities and Applications in Counting and Optimization

A polynomial p∈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 z1z2dots zn 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.