Tractable Set Constraints

TL;DR

This paper introduces a broad class of set constraint satisfaction problems (CSPs) called EI, which can be solved efficiently in quadratic time, extending previous tractable classes and impacting areas like description logics.

Contribution

The paper defines the EI class of set CSPs, proves it is solvable in quadratic time, and provides a universal-algebraic characterization showing larger classes are NP-hard.

Findings

EI set constraints are solvable in quadratic time

EI includes all previously known tractable set CSPs

Larger classes containing EI are NP-hard

Abstract

Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tractable. We introduce a large class of set CSPs that can be solved in quadratic time. Our class, which we call EI, contains all previously known tractable set CSPs, but also some new ones that are of crucial importance for example in description logics. The class of EI set constraints has an elegant universal-algebraic characterization, which we use to show that every set constraint language that properly contains all EI set constraints already has a finite sublanguage with an NP-hard constraint satisfaction problem.