Bounding and Approximating Kostka Numbers

Abstract

Kostka numbers, the coefficients of Schur polynomials, count semistandard Young tableaux of prescribed shape and weight. We give a deterministic algorithm for approximating Kostka numbers within a simply exponential factor in the number of variables and the size of the partition. Our approach combines Gurvits capacity bounds for log-concave polynomials with an algorithm for approximating the Gurvits capacity of a Schur polynomial. For a Schur polynomial in $n$ variables indexed by $\lambda$, we show that the logarithm of the capacity associated with any nonzero coefficient can be approximated to additive error $\epsilon$ in time polynomial in $n$, $|\lambda |$, and $\log (1/\epsilon )$. The analysis establishes an explicit bounded optimization region and efficient approximate first-order oracles, and handles boundary weights through a factorization into smaller Schur polynomials.