arXiv 2602.059572026-02-26math.CO

非负整数秩为二的矩阵

Matrices of nonnegative integer rank two

João Gouveia, Amy Wiebe

AI总结：

研究非负整数秩为二的矩阵，通过几何方法和算法解决rank2问题，揭示小矩阵对问题难度的影响。

摘要：

非负整数秩是经典非负秩的一种变体，于20世纪80年代提出，其中因数分解需要具有整数条目。尽管计算非负整数秩通常非常困难，我们专注于一个基本的特殊情况：确定秩为2的非负整数矩阵是否具有非负整数秩等于2（即“rank2问题”）。尽管在连续情况下这个问题很简单，但在这种背景下却出人意料地丰富。我们提供了一种几何改写形式，用平面中的仿射半群和有理锥来表示，从而获得新的结构见解。我们证明了任何秩为2的整数矩阵都可以被缩减为一个3×3矩阵，该矩阵具有非负整数秩2当且仅当原始矩阵也具有非负整数秩2，且这种缩减可以在多项式时间内计算。这种缩减揭示了rank2问题的难度已经由小矩阵所捕捉。基于这一几何框架，我们还开发了一种算法，通过在相关锥的有限区域内战略性地搜索整数生成器来解决rank2问题。尽管理论上的最坏情况复杂性仍然很高，但数值实验表明该算法在实践中表现高效。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文