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Tropically Convex Constraint Satisfaction

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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 P ∩ c 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 P ∩ c 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.

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Notes

  1. Interpretations in the sense of model theory; we refer to Hodges [20] since we do not need this concept further.

  2. The original definition of tropical convexity is for the dual situation, considering min instead of max .

  3. Also the results in Jeavons and Cooper [21] have been formulated in the dual situation for min instead of max .

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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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Authors and Affiliations

  1. Institut für Algebra, TU Dresden, 01062, Dresden, Germany

    Manuel Bodirsky & Marcello Mamino

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  1. Manuel Bodirsky
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  2. Marcello Mamino
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Correspondence to Marcello Mamino.

Additional information

An extended abstract appeared in the proceedings of the 11th International Computer Science Symposium in Russia, CSR 2016.

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Bodirsky, M., Mamino, M. Tropically Convex Constraint Satisfaction. Theory Comput Syst 62, 481–509 (2018). https://doi.org/10.1007/s00224-017-9762-0

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