Capacity Lower Bounds via Productization

TL;DR

This paper introduces a new technique called productization to establish sharp lower bounds on the capacity of real stable polynomials, improving previous bounds and enabling advances in approximation algorithms for problems like metric TSP.

Contribution

The paper develops the productization technique to derive improved capacity lower bounds for real stable polynomials, independent of degree and with exponential dependence on variables.

Findings

Sharp lower bound on capacity depending on gradient at 1

Improved bounds for non-homogeneous polynomials with exponential dependence

Application to new scaling algorithms and approximation improvements

Abstract

We give a sharp lower bound on the capacity of a real stable polynomial, depending only on the value of its gradient at $x = 1$. This result implies a sharp improvement to a similar inequality proved by Linial-Samorodnitsky-Wigderson in 2000, which was crucial to the analysis of their permanent approximation algorithm. Such inequalities have played an important role in the recent work on operator scaling and its generalizations and applications, and in fact we use our bound to construct a new scaling algorithm for real stable polynomials. In addition, we give a strong improvement on previous lower bounds of the capacity of a non-homogeneous real stable polynomial, depending only on the value of its gradient at $x = 1$. Crucially, this new bound is independent of the degree of the polynomial, and has singly exponential dependence on the number of variables. This compares favorably to the bounds used recently in the fantastic work of Karlin-Klein-Oveis Gharan to give an improved approximation factor for metric TSP, where this dependence is doubly exponential. Such bounds were conjectured to exist by the authors, and thus our new bound should imply further improvement to the approximation factor for metric TSP. The new technique we develop to prove this bound is productization, which says that any real stable polynomial can be approximated at any point in the positive orthant by a product of linear forms. Beyond the results of this paper, our main hope is that this new technique will allow us to avoid "frightening technicalities", in the words of Laurent and Schrijver, that often accompany combinatorial lower bounds.