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

Yuki Nishida

TL;DR

This work addresses the CSR expansion for max-plus matrices and introduces an $O(n(m+n\log n))$ algorithm that leverages the roots of the characteristic polynomial $\chi_A(t)$ as growth rates in the CSR terms. The core theoretical contribution is that each growth rate $\lambda_k$ in the CSR expansion is a root of $\chi_A(t)$, enabling a decomposition guided by the maximal multi-circuit sequence (MMCS). The method combines the Gassner–Klinz root-finding for $\chi_A(t)$ with graph-visualization and a graph-extension technique to efficiently compute the CSR components $C_k$, $S_k$, and $R_k$, improving on the previous $O(n^4\log n)$ algorithm. An illustrative example demonstrates the workflow and confirms the identified growth rates and CSR factors, highlighting the practical impact for long-horizon max-plus dynamics and discrete-event systems.

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.

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

TL;DR

This work addresses the CSR expansion for max-plus matrices and introduces an algorithm that leverages the roots of the characteristic polynomial as growth rates in the CSR terms. The core theoretical contribution is that each growth rate in the CSR expansion is a root of , enabling a decomposition guided by the maximal multi-circuit sequence (MMCS). The method combines the Gassner–Klinz root-finding for with graph-visualization and a graph-extension technique to efficiently compute the CSR components , , and , improving on the previous algorithm. An illustrative example demonstrates the workflow and confirms the identified growth rates and CSR factors, highlighting the practical impact for long-horizon max-plus dynamics and discrete-event systems.

Abstract

Max-plus algebra is a semiring with addition and multiplication . 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 -by- matrix admits the CSR expansion that decomposes the matrix into a sum of at most periodic terms after times of powers. In this study, we propose an time algorithm for the CSR expansion, where is the number of nonzero entries in the matrix, which improves the 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.
Paper Structure (11 sections, 13 theorems, 50 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 13 theorems, 50 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries on max-plus algebra
  3. Max-plus matrices and graphs
  4. Eigenvalues and eigenvectors
  5. CSR decomposition
  6. Our algorithm
  7. Decomposition based on roots of characteristic polynomials
  8. Visualization of matrices
  9. Computing the CSR decomposition
  10. Example
  11. Concluding remarks

Key Result

Proposition 2.1

For a matrix $A \in \mathbb{R}_{\max}^{n \times n}$, the maximum mean weight of all elementary circuits in $\mathcal{G}(A)$ is the maximum eigenvalue of $A$.

Figures (2)

  • Figure 1: Graph $\mathcal{G}(A)$ in Section 4.4. Red-bold arrows represent quasi-critical circuits.
  • Figure 2: Extended graph $\hat{G}$. The weights of the arcs are $0$ if they are not specified.

Theorems & Definitions (14)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3: Green1983, Theorem 3
  • Proposition 2.4: Gassner2009
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 4.5
  • Lemma 4.6
  • ...and 4 more