On tropical and nonnegative factorization ranks of band matrices

TL;DR

This paper surveys the computational complexity of various matrix factorizations over different semirings, focusing on band matrices, and identifies cases where these problems become tractable.

Contribution

It demonstrates that certain matrix factorization problems over semirings are polynomial-time solvable for band matrices and introduces new open problems.

Findings

Boolean, fuzzy, and tropical factorizations are polynomial-time solvable for band matrices.

Nonnegative rank of tridiagonal matrices can be computed efficiently.

The paper highlights open problems and directions for future research.

Abstract

Matrix factorization problems over various semirings naturally arise in different contexts of modern pure and applied mathematics. These problems are very hard in general and cause computational difficulties in applications. We give a survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and positive semidefinite factorizations, and we examine the behavior of the corresponding rank functions on matrices of bounded bandwidth. We show that the Boolean, fuzzy, and tropical versions of matrix factorization become polynomial time solvable when restricted to this class of matrices, and we also show that the nonnegative rank of a tridiagonal matrix is easy to compute. We recall several open problems from earlier papers on the topic and formulate many new problems.