Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On max-plus two-sided linear systems whose solution sets are min-plus linear

Yasutaka Ooga, Yuki Nishida, Yoshihide Watanabe

TL;DR

This work addresses the problem of characterizing all solutions to the max-plus two-sided linear system $A \otimes \mathbf{x} = B \otimes \mathbf{x}$. It leverages the alternating method, traditionally used for separated two-sided systems, to generate a set of candidate solutions and shows that the min-plus linear closure of these solutions contains the entire solution set; under projective boundedness, this yields a finite, min-plus representation of the solution set. A key contribution is a sufficient condition for the solution set to be min-plus linear, based on local min-plus convexity around vectors produced by the alternating method. The results establish pseudo-polynomial complexity bounds and illuminate the connection between max-plus and min-plus structures, with implications for tropical linear systems and related mean payoff game formulations.

Abstract

The max-plus algebra $\mathbb{R}\cup \{-\infty \}$ is defined in terms of a combination of the following two operations: addition, $a \oplus b := \max(a,b)$, and multiplication, $a \otimes b := a + b$. In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system $A \otimes x = B \otimes x$. We demonstrate that the minimum ``min-plus'' linear subspace containing the ``max-plus'' solution space can be computed by applying the alternating method algorithm, which is a well-known method to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the ``min-plus'' and ``max-plus'' subspaces to be identical. The computational complexity of the method presented in this study is pseudo-polynomial.

On max-plus two-sided linear systems whose solution sets are min-plus linear

TL;DR

This work addresses the problem of characterizing all solutions to the max-plus two-sided linear system . It leverages the alternating method, traditionally used for separated two-sided systems, to generate a set of candidate solutions and shows that the min-plus linear closure of these solutions contains the entire solution set; under projective boundedness, this yields a finite, min-plus representation of the solution set. A key contribution is a sufficient condition for the solution set to be min-plus linear, based on local min-plus convexity around vectors produced by the alternating method. The results establish pseudo-polynomial complexity bounds and illuminate the connection between max-plus and min-plus structures, with implications for tropical linear systems and related mean payoff game formulations.

Abstract

The max-plus algebra is defined in terms of a combination of the following two operations: addition, , and multiplication, . In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system . We demonstrate that the minimum ``min-plus'' linear subspace containing the ``max-plus'' solution space can be computed by applying the alternating method algorithm, which is a well-known method to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the ``min-plus'' and ``max-plus'' subspaces to be identical. The computational complexity of the method presented in this study is pseudo-polynomial.
Paper Structure (11 sections, 15 theorems, 75 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 15 theorems, 75 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Max-plus algebra
  3. Max-plus and min-plus algebras
  4. Max-plus linear algebra
  5. Max-plus linear systems
  6. Alternating method for solving two-sided systems
  7. Stable solutions obtained via the alternating method
  8. Description of solutions as min-plus linear subspaces
  9. Projectively bounded solution spaces
  10. Min-plus linear closure of the solution set
  11. A criterion for the solution set to be min-plus linear

Key Result

Proposition 3.2

If $A\in\mathbb{Z}^{m\times n},B\in(\mathbb{Z}\cup\{\varepsilon\})^{m\times k}$ and $\bm{x}(0) \in\mathbb{Z}^n$, then the alternating method terminates after finitely many steps. Further, if $|x_{i}(0)|$ is bounded by $K:=\max_{i,j} |a_{ij}|$, the computational complexity is $O(mn(n+k)K)$.

Figures (2)

  • Figure 1: The solution set, $S(A,B)$, projected onto $x_{1}=0$ (lattice pattern) and its min-plus linear closure (shaded).
  • Figure 2: Proof of Lemma \ref{['loc-glob']}.

Theorems & Definitions (20)

  • Proposition 3.2: Green2003
  • Example 3.3
  • Lemma 3.4: Green2003
  • Lemma 3.5: Green2003
  • Proposition 3.6
  • Lemma 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Corollary 4.5
  • ...and 10 more