The complexity of tropical matrix factorization

TL;DR

This paper proves that the problem of tropical matrix factorization is computationally hard, indicating no polynomial-time algorithm exists for all fixed sizes, which resolves a long-standing open problem.

Contribution

It establishes the computational complexity of tropical matrix factorization, showing the problem is unlikely to be solvable in polynomial time for fixed parameters.

Findings

TMF is computationally hard for fixed k

No polynomial-time algorithm likely exists for general TMF

Resolves a problem posed by Barvinok in 1993

Abstract

The tropical arithmetic operations on $\mathbb{R}$ are defined by $a\oplus b=\min\{a,b\}$ and $a\otimes b=a+b$. Let $A$ be a tropical matrix and $k$ a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices $B\in\mathbb{R}^{m\times k}$ and $C\in\mathbb{R}^{k\times n}$ satisfying $B\otimes C=A$. We show that no algorithm for TMF is likely to work in polynomial time for every fixed $k$, thus resolving a problem proposed by Barvinok in 1993.