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. 
Damian Straszak and Nisheeth K. Vishnoi
cs.DS math.CO math.OC math.PR