Max-Closed Semilinear Constraint Satisfaction

  • Manuel Bodirsky
  • Marcello MaminoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


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 \(\mathsf {NP}\cap \mathsf {co}\text {-}\mathsf {NP}\), 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 \(\mathsf {NP}\cap \mathsf {co}\text {-}\mathsf {NP}\). 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 \(\mathsf {P}\); 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 \(\mathsf {NP}\)-hard.



Both authors have received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039), and the German Research Foundation (DFG, project number 622397).


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für AlgebraTU DresdenDresdenGermany

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