Deterministically approximating the volume of a Kostka polytope
Hariharan Narayanan, Piyush Srivastava
TL;DR
The paper tackles the problem of deterministically approximating the volume of Kostka polytopes GT(λ, μ), a question tied to random matrix theory through the randomized Schur-Horn problem. The authors introduce a continuous analogue of Schur polynomials, S_λ(x), interpreted as a Gibbs partition function, and use it to obtain a rigorous lower bound on vol(GT(λ, μ)); they also derive an upper bound by exploiting geometric conditioning and ball containment within GT(λ, μ). The core result bounds vol(GT(λ, μ)) relative to an infimum of S_λ(x) exp(- x^T μ) and provides a polynomial-time algorithm to compute a (1±ε) approximation of that infimum under the regime where λ is an integral partition with n parts (entries of λ bounded by poly(n)) and μ lies in the interior of SH(λ). The approach reduces the volume estimation to convex optimization with a determinant-based evaluation oracle for S_λ and carefully handles numerical representation lengths, yielding a sub-exponential overall approximation factor in the dimension. This work advances deterministic volume approximation for a natural representation-theoretic polytope class and connects geometric, combinatorial, and optimization techniques with implications for Kostka numbers and related marginal problems.
Abstract
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 $λ$ has a given diagonal $μ$ (the so-called randomized Schur-Horn problem): the corresponding Kostka polytope is denoted $\mathrm{GT}(λ, μ)$. We give a polynomial-time deterministic algorithm for approximating the volume of a ($Ω(n^2)$ dimensional) Kostka polytope $\mathrm{GT}(λ, μ)$ to within a multiplicative factor of $\exp(O(n\log n))$, when $λ$ is an integral partition with $n$ parts, with entries bounded above by a polynomial in $n$, and $μ$ is an integer vector lying in the interior of the permutohedron (i.e., convex hull of all permutations) of $λ$. The algorithm thus gives asymptotically correct estimates of the log-volume of Kostka polytopes corresponding to such $(λ, μ)$. Our approach is based on a partition function interpretation of a continuous analogue of Schur polynomials.