Matrices of nonnegative integer rank two

Abstract

The nonnegative integer rank of a matrix is a variant of the classical nonnegative rank, introduced in the 1980s, where factorizations are required to have integer entries. While computing nonnegative integer rank is generally very hard, we focus on a fundamental special case: determining when a rank 2 nonnegative integer matrix has nonnegative integer rank equal to 2 (the "rank2 problem"). Although this problem is trivial in the continuous case, in this context it is surprisingly rich. We provide a geometric reformulation in terms of affine semigroups and rational cones in the plane, which yields new structural insights. We show that any rank 2 integer matrix can be reduced to a $3\times 3$ matrix which has nonnegative integer rank $2$ if and only if the original one also has nonnegative integer rank $2$, with the reduction computable in polynomial time. This reduction reveals that the difficulty of the rank2 problem is already captured by small matrices. Building on this geometric framework, we also develop an algorithm that solves the rank2 problem by strategically searching for integer generators within bounded regions of the associated cone. Although the theoretical worst-case complexity remains high, numerical experiments demonstrate that the algorithm performs efficiently in practice.