An order out of nowhere: a new algorithm for infinite-domain CSPs

TL;DR

This paper introduces a novel algorithmic approach for infinite-domain CSPs, establishing a complexity dichotomy and confirming a conjecture for a broad class of hypergraph-based problems.

Contribution

It presents a new algorithm inspired by finite-domain CSP techniques and proves a P/NP-complete dichotomy for infinite hypergraph CSPs, extending previous graph satisfiability results.

Findings

Established a P/NP-complete complexity dichotomy for many infinite hypergraph CSPs.

Introduced an algorithmic technique based on symmetries and linear orders.

Confirmed the Bodirsky-Pinsker conjecture for various homogeneous hypergraphs.

Abstract

We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of many homogeneous hypergraphs including the random hypergraphs and hypergraphs omitting a generalised clique. This forms a vast generalisation of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongr\'acz (ICALP'16) on graph satisfiability.