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Real Stable Polynomials and Matroids: Optimization and Counting

A great variety of 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 $m$-variate real polynomial $g$ and a family of subsets $B$ of $[m]$, (1) find $S\in B$ such that the monomial in $g$ corresponding to $S$ has the largest coefficient in $g$, or (2) compute the sum of coefficients of monomials in $g$ corresponding to all the sets in $B$. Special cases of these problems, such as computing permanents, sampling from DPPs and maximizing subdeterminants have been topics of recent interest in theoretical computer science. In this paper we present a general convex programming framework geared to solve both of these problems. 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 relaxation is finite and depends only on $m$ (and not on the coefficients of g). Prior to our work, such results were known only in sporadic cases that relied on the structure of $g$ and $B$; it was not even 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 on the van der Waerden conjecture for real stable $g$ when $B$ is a single element and a result by Nikolov and Singh for multilinear real stable polynomials when $B$ is a partition matroid. Our work, which encapsulates most interesting cases of $g$ and $B$, benefits from both - we were inspired by the latter in deriving the right convex programming relaxation and the former in establishing the integrality gap. However, proving our results requires significant extensions of both; in that process we come up with new notions and connections between stable polynomials and matroids which should be of independent interest.