On the Column Number and Forbidden Submatrices for Δ-modular Matrices

An integer matrix A is A-modular if the determinant of each rank(A) \times rank(A) submatrix of A has absolute value at most A. The study of A-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by A-modular constraint matrices can be solved in polynomial time if A is considered constant. The conjecture is known to hold true only when A \in {1, 2}. In light of this conjecture, a natural question is to understand structural properties of A-modular matrices. We consider the column number question, how many nonzero, pairwise nonparallel columns can a rank-r A-modular matrix have? We prove that for each positive integer A and sufficiently large integer r, every rank-r A-modular matrix has at most (r+1\bigr) + 80A7 \cdot r nonzero, pairwise nonparallel columns, which is tight up to the term 80A7. This is the first upper bound of the form (r +1 2 \bigr) + f (A) \cdot r with f a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a A-modular matrix. We believe this partial list may be of independent interest in future studies of A-modular matrices.