Max-Closed Semilinear Constraint Satisfaction

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 ini¾ź$$\mathsf {NP}\cap \mathsf {co}\text {-}\mathsf {NP}$$, 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 intoi¾ź$$\mathsf {NP}\cap \mathsf {co}\text {-}\mathsf {NP}$$. 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 toi¾źL, the CSP becomes $$\mathsf {NP}$$-hard.