Tropically convex constraint satisfaction

TL;DR

This paper studies the complexity of constraint satisfaction problems involving tropically convex semilinear relations, establishing new duality results and complexity classifications that extend known results in scheduling and linear inequalities.

Contribution

It introduces a new duality for open tropically convex relations, placing the CSP for these constraints in NP intersected co-NP, and characterizes max-closed semilinear relations via logical and relational frameworks.

Findings

CSP for tropically convex semilinear constraints is in NP ∩ co-NP.

A subclass of max-closed constraints has polynomial-time solvability.

Adding any non-max-closed relation makes the CSP NP-hard.

Abstract

A semilinear relation S 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 the intersection of NP and co-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 semilinaer constraints in general into NP intersected co-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 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 NP-hard.