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Discrete Event Dynamic Systems

, Volume 29, Issue 2, pp 163–189 | Cite as

Sparsity in max-plus algebra and systems

  • Anastasios TsiamisEmail author
  • Petros Maragos
Article

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 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.

Keywords

Max-plus algebra Max-plus systems Sparsity Supermodularity 

Notes

Acknowledgments

P. Maragos’ work was partially supported by the European Regional Development Fund of the EU and Greek national funds through the Operational Program Competitiveness, Entrepreneurship and Innovation, under the call ‘Research - Create - Innovate’ (T1EDK-01248, “i-Walk”).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical and Systems EngineeringUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.School of Electrical and Computer EngineeringNational Technical University of AthensAthensGreece

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