On the size of integer programs

We are interested in a combinatorial question related to the ongoing interest in understanding the complexity of integer linear programming with bounded subdeterminants.

Given a positive integer Delta and a full-rank integer matrix A with m rows such that the absolute value of every m-by-m minor of A is bounded by Delta, at most how many pairwise distinct columns can A have? The case Delta = 1 is the classical result of Heller (1957) saying that the maximal number of pairwise distinct columns of a totally unimodular integer matrix with m rows is m^2 + m + 1. If m >= 3 and Delta >= 3, the precise answer to this combinatorial question is not known, but bounds of order O(Delta^2 m^2) have been proven by Lee et al. (2021). In the talk, I will discuss our approach to the problem which rests on counting columns of A by residue classes of a suitably defined integer lattice. As a result, we obtain the first bound that is linear in Delta and polynomial (of low degree) in the dimension m.

This is joint work with Gennadiy Averkov.