On the Recognition of a, b, c-Modular Matrices

Christoph Glanzer, Ingo Stallknecht, Robert Weismantel

Abstract: Let $$A \in \mathbb {Z}^{m \times n}$$ be an integral matrix and $$a, b, c \in \mathbb {Z}$$ satisfy $$a \ge b \ge c \ge 0$$ . The question is to recognize whether A is $$\{a,b,c\}$$ -modular, i.e., whether the set of $$n \times n$$ 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 $$n \times n$$ subdeterminants $$k_1$$ and $$k_2$$ satisfying $$2 \cdot |k_1| = |k_2|$$ . 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.