Real stable polynomials and matroids: optimization and counting

Several fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an oracle access to an m-variate real polynomial g and to a family of (multi-)subsets ℬ of [m], (1) compute the sum of coefficients of monomials in g corresponding to all the sets that appear in B(1), or find S ε ℬ such that the monomial in g corresponding to S has the largest coefficient in g. Special cases of these problems, such as computing permanents and mixed discriminants, sampling from determinantal point processes, and maximizing sub-determinants with combinatorial constraints have been topics of much recent interest in theoretical computer science. In this paper we present a general convex programming framework geared to solve both of these problems. Subsequently, we show that roughly, when g is a real stable polynomial with non-negative coefficients and B is a matroid, the integrality gap of our convex relaxation is finite and depends only on m (and not on the coefficients of g). Prior to this work, such results were known only in important but sporadic cases that relied heavily on the structure of either g or ℬ; it was not even a priori clear if one could formulate a convex relaxation that has a finite integrality gap beyond these special cases. Two notable examples are a result by Gurvits for real stable polynomials g when ℬ contains one element, and a result by Nikolov and Singh for a family of multi-linear real stable polynomials when B is the partition matroid. This work, which encapsulates almost all interesting cases of g and B, benefits from both - it is inspired by the latter in coming up with the right convex programming relaxation and the former in deriving the integrality gap. However, proving our results requires extensions of both; in that process we come up with new notions and connections between real stable polynomials and matroids which might be of independent interest.