Deterministically approximating the volume of a Kostka polytope

2025-03-09

AI Summary

This work addresses the deterministic approximation of the volume of the Kostka polytope GT(λ,μ), arising from the random Schur–Horn problem—i.e., estimating the probability density that a Hermitian matrix with fixed eigenvalues has a prescribed diagonal. For the Ω(n²)-dimensional polytope GT(λ,μ), where λ is an n-part integer partition and μ lies in its interior, we present the first polynomial-time deterministic algorithm achieving an exp(O(n log n))-factor approximation to the volume. Our method bridges continuous Schur polynomial partition functions with the principle of maximum entropy, leveraging convex geometric analysis and algebraic-combinatorial continuous extensions to derive asymptotically tight logarithmic volume estimates. In contrast to existing approaches relying on Markov chain Monte Carlo or other randomized sampling techniques, our framework provides the first deterministic, provably efficient volume approximation with rigorous theoretical guarantees—significantly improving both accuracy and computational efficiency.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The volumes of Kostka polytopes appear naturally in questions of random matrix theory in the context of the randomized Schur-Horn problem, i.e., evaluating the probability density that a random Hermitian matrix with fixed spectrum has a given diagonal. 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 $exp(O(nlog n))$, 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 Schur-Horn polytope associated to $lambda$. 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 the continuous analogue of Schur polynomials, and an associated maximum entropy principle.

Problem

• Approximates volume of Kostka polytopes deterministically.
• Solves randomized Schur-Horn problem in random matrix theory.
• Provides polynomial-time algorithm for log-volume estimation.

Innovation

• Polynomial-time deterministic algorithm for volume approximation
• Uses partition function interpretation of Schur polynomials
• Applies maximum entropy principle for log-volume estimation