Notes on {a,b,c}-Modular Matrices

Abstract

Let A in mathbb {Z}^{m times n} be an integral matrix and a, b, c in mathbb {Z} satisfy a ≥ b ≥ c ≥ 0. The question is to recognize whether A is {a,b,c}-modular, i.e., whether the set of n × n subdeterminants of A in absolute value is {a,b,c}. We will succeed in solving this problem in polynomial time unless A possesses a duplicative relation, that is, A has nonzero n × n subdeterminants k1 and k2 satisfying 2 ⋅|k1| = |k2|. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over {a,b,c}-modular constraint matrices for any constants a, b and c.