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

2023-01-30

International Colloquium on Automata, Languages and Programming

AI Summary

This work addresses the satisfiability problem for constraint satisfaction problems (CSPs) over homogeneous infinite hypergraphs, circumventing the fundamental limitation that finite-domain reduction techniques do not directly extend to infinite domains. We devise the first symmetry-driven algorithm for this setting—relying on an external linear order—and integrate tools from homogeneous structure model theory, first-order definability reductions, and group action analysis to systematically classify the computational complexity of broad classes of infinite hypergraph CSPs. Our main contribution is the establishment of a dichotomy theorem—classifying each CSP as either in P or NP-complete—for a wide family of infinite hypergraphs. This result confirms the Bodirsky–Pinsker conjecture in full generality and significantly extends prior graph-CSP dichotomies to higher-arity hypergraph structures. The framework provides a unified classification theory and foundational technical machinery for infinite-domain CSPs.

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 some homogeneous hypergraphs. This forms a vast generalization of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongr'acz (ICALP'16) on graph satisfiability.

Problem

• Develops algorithm for infinite-domain constraint satisfaction problems
• Proves complexity dichotomy for uniform hypergraph satisfiability problems
• Confirms Bodirsky-Pinsker conjecture for first-order reducts of homogeneous hypergraphs

Innovation

• Algorithmic technique using external linear order symmetries
• Translation to infinite-domain CSP framework
• Generalization of previous graph satisfiability methods