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Matrices of nonnegative integer rank two

João Gouveia, Amy Wiebe

TL;DR

This work studies the nonnegative integer rank and, in particular, the rank-$2$ problem, by recasting it as a geometric question on affine semigroups and rational planar cones. A polynomial-time reduction shows any rank-2 instance can be transformed to a $3\times3$ matrix with the same property, enabling a bounded-search algorithm that exploits a cone-decomposition structure. The proposed algorithm constructs canonical diagrams via Smith Normal Form, decomposes the associated cone, bounds the search region, and recovers an explicit factorization if possible; numerical tests confirm practical efficiency, especially when using the $3\times3$ reduction for large inputs. The framework clarifies the problem’s geometry and opens routes for extensions, while acknowledging fundamental hardness and limitations beyond rank $2$.

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.

Matrices of nonnegative integer rank two

TL;DR

This work studies the nonnegative integer rank and, in particular, the rank- problem, by recasting it as a geometric question on affine semigroups and rational planar cones. A polynomial-time reduction shows any rank-2 instance can be transformed to a matrix with the same property, enabling a bounded-search algorithm that exploits a cone-decomposition structure. The proposed algorithm constructs canonical diagrams via Smith Normal Form, decomposes the associated cone, bounds the search region, and recovers an explicit factorization if possible; numerical tests confirm practical efficiency, especially when using the reduction for large inputs. The framework clarifies the problem’s geometry and opens routes for extensions, while acknowledging fundamental hardness and limitations beyond rank .

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 matrix which has nonnegative integer rank if and only if the original one also has nonnegative integer rank , 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.
Paper Structure (12 sections, 3 theorems, 25 equations, 10 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 3 theorems, 25 equations, 10 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A geometric view of the rank2 problem
  3. A geometric reformulation
  4. Reduction to the $3\times 3$ case
  5. Algorithm
  6. Algorithm details
  7. Constructing the diagram
  8. Decomposing the Cone
  9. Bounding the Search Space
  10. Factorization from generators
  11. Numerical Testing
  12. Conclusion

Key Result

Proposition 2.8

For a rational pointed cone with nonempty interior $K \subseteq \mathbb{R}^2$ and $m$ integer points $a_1,...,a_m \in K$, there is a $3 \times m$ matrix $B$ such that $\mathop{\mathrm{rank_{\mathbb{Z}}^+}}\nolimits(B)=2$ if and only if there are two integer points in $K$ such that every $a_i$ is a n

Figures (10)

  • Figure 1: Graphical interpretation of Example \ref{['ex:graphicexample']}.
  • Figure 2: Graphical interpretation of Example \ref{['ex:graphicexample2']}.
  • Figure 3: Transformation of Figure \ref{['fig:graphicexample2']} to a canonical form.
  • Figure 4: Diagrams of a $5 \times 5$ matrix and its reduction.
  • Figure 5: Diagram of a $3 \times 4$ matrix.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Example 2.3
  • Example 2.4
  • Example 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • Lemma 2.9
  • proof
  • Theorem 2.10
  • Remark 2.11
  • ...and 4 more