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.

翻译：多项式时间内确定性近似计算多面体体积（近似因子至多随维度次指数增长）仍是一个悬而未决的问题。该问题的最新研究聚焦于识别能够获得此类近似算法的有意义多面体类别。本文重点研究其中一类多面体：Kostka多面体。Kostka多面体的体积自然出现在随机矩阵理论问题中，具体涉及计算具有固定谱$\lambda$的随机埃尔米特矩阵具有给定对角线$\mu$的概率密度（即随机化Schur-Horn问题）：对应的Kostka多面体记为$\mathrm{GT}(\lambda, \mu)$。当$\lambda$为具有$n$个分量的整数分割且其分量值以$n$的多项式为上界，且$\mu$为位于$\lambda$的置换体（即所有排列的凸包）内部的整数向量时，我们提出一种多项式时间确定性算法，能以$\exp(O(n\log n))$的乘性因子近似计算（$\Omega(n^2)$维）Kostka多面体$\mathrm{GT}(\lambda, \mu)$的体积。该算法因此能对此类$(\lambda, \mu)$对应的Kostka多面体对数体积给出渐近正确的估计。我们的方法基于Schur多项式连续模拟的配分函数解释。