Tractable hypergraph properties for constraint satisfaction and conjunctive queries

TL;DR

This paper introduces a new hypergraph measure called submodular width, which characterizes the fixed-parameter tractability of constraint satisfaction problems based on hypergraph structure, extending understanding beyond treewidth.

Contribution

The paper defines submodular width as a new hypergraph property and proves it determines the fixed-parameter tractability of CSPs, generalizing previous results based on fractional hypertree width.

Findings

Bounded submodular width implies fixed-parameter tractability of CSP(H).

Unbounded submodular width leads to non-fixed-parameter tractability unless ETH fails.

Introduces a new measure that extends the understanding of hypergraph properties in CSP complexity.

Abstract

An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomial-time solvable or fixed-parameter tractable, parameterized by the number of variables. Note that in the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameterization by the number of variables is a meaningful question. The most general known property of H that makes CSP(H) polynomial-time solvable is bounded fractional hypertree width. Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H implies that CSP(H) is fixed-parameter tractable. In a matching hardness result, we show that if H has unbounded submodular width, then CSP(H) is not fixed-parameter tractable, unless the Exponential Time Hypothesis fails.