  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.
