Sparsity in Max-Plus Algebra and Systems

TL;DR

This paper investigates sparsity in max-plus algebra, establishing complexity results, proposing greedy approximation algorithms with guarantees, and demonstrating applications in resource optimization and system identification.

Contribution

It introduces a framework for sparse solutions in max-plus algebra, analyzes their complexity, and develops polynomial-time greedy algorithms with proven bounds for practical problems.

Findings

Sparsest max-plus solutions are NP-complete to find.

Greedy algorithms provide approximate solutions with guaranteed bounds.

Applications include resource optimization and system identification.

Abstract

We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the $\ell_{1}$ residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations.