Deterministically approximating the volume of a Kostka polytope

Polynomial-time deterministic approximation of volumes of polytopes, up to an approximation factor that grows at most sub-exponentially with the dimension, remains an open problem. Recent work on this question has focused on identifying interesting classes of polytopes for which such approximation algorithms can be obtained. In this paper, we focus on one such class of polytopes: the Kostka polytopes. The volumes of Kostka polytopes appear naturally in questions of random matrix theory, in the context of evaluating the probability density that a random Hermitian matrix with fixed spectrum $\lambda$ has a given diagonal $\mu$ (the so-called randomized Schur-Horn problem): the corresponding Kostka polytope is denoted $\mathrm{GT}(\lambda, \mu)$. We give a polynomial-time deterministic algorithm for approximating the volume of a ($\Omega(n^2)$ dimensional) Kostka polytope $\mathrm{GT}(\lambda, \mu)$ to within a multiplicative factor of $\exp(O(n\log n))$, when $\lambda$ is an integral partition with $n$ parts, with entries bounded above by a polynomial in $n$, and $\mu$ is an integer vector lying in the interior of the permutohedron (i.e., convex hull of all permutations) of $\lambda$. The algorithm thus gives asymptotically correct estimates of the log-volume of Kostka polytopes corresponding to such $(\lambda, \mu)$. Our approach is based on a partition function interpretation of a continuous analogue of Schur polynomials.