Capacity Lower Bounds via Productization

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. Our bound is also quite similar to one used very recently by Karlin-Klein-Oveis Gharan to give an improved 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. In particular, we believe that this technique will be useful towards simplifying and improving further the approximation factor given in the fantastic work of Karlin-Klein-Oveis Gharan on metric TSP.Comment: STOC 202