Algorithm for the CSR expansion of max-plus matrices using the characteristic polynomial

TL;DR

This paper introduces a new efficient algorithm for the CSR expansion of max-plus matrices by leveraging the roots of the characteristic polynomial, significantly reducing computational complexity.

Contribution

It presents an $O(n(m+n \log n))$ time algorithm for CSR expansion, improving upon the previous $O(n^{4} \log n)$ method, based on characteristic polynomial roots.

Findings

The algorithm efficiently computes the CSR expansion for max-plus matrices.

It reduces the computational complexity from $O(n^{4} \log n)$ to $O(n(m+n \log n))$.

The roots of the characteristic polynomial serve as growth rates in the CSR expansion.

Abstract

Max-plus algebra is a semiring with addition $a\oplus b = \max(a,b)$ and multiplication $a\otimes b = a+b$. It is applied in cases, such as combinatorial optimization and discrete event systems. We consider the power of max-plus square matrices, which is equivalent to obtaining the all-pair maximum weight paths with a fixed length in the corresponding weighted digraph. Each $n$-by-$n$ matrix admits the CSR expansion that decomposes the matrix into a sum of at most $n$ periodic terms after $O(n^{2})$ times of powers. In this study, we propose an $O(n(m+n \log n))$ time algorithm for the CSR expansion, where $m$ is the number of nonzero entries in the matrix, which improves the $O(n^{4} \log n)$ algorithm known for this problem. Our algorithm is based on finding the roots of the characteristic polynomial of the max-plus matrix. These roots play a similar role to the eigenvalues of the matrix, and become the growth rates of the terms in the CSR expansion.