Semantic Width and the Fixed-Parameter Tractability of Constraint Satisfaction Problems

TL;DR

This paper characterizes when certain constraint satisfaction problems (CSPs) are fixed-parameter tractable, extending understanding to unions of conjunctive queries and revealing new insights into the complexity boundaries of CSPs.

Contribution

It provides a novel characterization of fixed-parameter tractability for CSPs and their unions, resolving a long-standing open problem and enhancing the CSP framework's utility.

Findings

Characterization of fixed-parameter tractability for CSPs.

Extension of results to unions of conjunctive queries.

Insight into the PTIME solvability frontier of CSPs.

Abstract

Constraint satisfaction problems (CSPs) are an important formal framework for the uniform treatment of various prominent AI tasks, e.g., coloring or scheduling problems. Solving CSPs is, in general, known to be NP-complete and fixed-parameter intractable when parameterized by their constraint scopes. We give a characterization of those classes of CSPs for which the problem becomes fixed-parameter tractable. Our characterization significantly increases the utility of the CSP framework by making it possible to decide the fixed-parameter tractability of problems via their CSP formulations. We further extend our characterization to the evaluation of unions of conjunctive queries, a fundamental problem in databases. Furthermore, we provide some new insight on the frontier of PTIME solvability of CSPs. In particular, we observe that bounded fractional hypertree width is more general than bounded hypertree width only for classes that exhibit a certain type of exponential growth. The presented work resolves a long-standing open problem and yields powerful new tools for complexity research in AI and database theory.