Constraint satisfaction problems for reducts of homogeneous graphs

TL;DR

This paper classifies the computational complexity of constraint satisfaction problems for structures definable in countably infinite homogeneous graphs, establishing a dichotomy between problems that are either polynomial-time solvable or NP-complete.

Contribution

It extends the complexity classification to all structures definable in homogeneous graphs, completing the dichotomy for these classes.

Findings

CSPs for structures in Henson graphs are either in P or NP-complete.

A similar dichotomy holds for structures definable in homogeneous graphs with an equivalence relation.

The classification unifies previous results, including those for the random graph.

Abstract

For $n\geq 3$, let $(H_n, E)$ denote the $n$-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on $n$ vertices. We show that for all structures $\Gamma$ with domain $H_n$ whose relations are first-order definable in $(H_n,E)$ the constraint satisfaction problem for $\Gamma$ is either in P or is NP-complete. We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation. Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.