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Theory of Computing Systems

, Volume 62, Issue 3, pp 481–509 | Cite as

Tropically Convex Constraint Satisfaction

  • Manuel Bodirsky
  • Marcello Mamino
Article
  • 71 Downloads

Abstract

A semilinear relation \(S \subseteq {\mathbb {Q}}^{n}\) is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in N Pc o N P, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinear constraints in general into N Pc o N P. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes N P-hard.

Keywords

Tropical convexity Semi-linear relations Max-closure Constraint satisfaction Max-plus-average inequalities Stochastic games Piecewise linear constraints Computational complexity 

Notes

Acknowledgments

Both authors have received funding from the European Research Council (grant agreement number 257039 and 681988), and from the German Research Foundation (DFG, project number 622397).

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut für AlgebraTU DresdenDresdenGermany

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