Asymptotic behavior of Integer Programming and the stability of the Castelnuovo-Mumford regularity

TL;DR

This paper explores the asymptotic behavior of certain integer programming problems and links these findings to the stability of Castelnuovo-Mumford regularity in commutative algebra, providing bounds and stability results.

Contribution

It establishes a connection between integer programming asymptotics and algebraic invariants, offering new bounds on the stability of Castelnuovo-Mumford regularity.

Findings

Integer programs have quasi-linear optima beyond a certain threshold.

Bounds are provided for the stability indices of algebraic regularities.

The results connect combinatorial optimization with algebraic geometry.

Abstract

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on $n$. An integer $N_*$ is determined such that the optima of these integer programs are a quasi-linear function of $n$ for all $n\ge N_*$. Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo-Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.