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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.

Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity