A Maximal Tractable Class of Soft Constraints

TL;DR

This paper identifies a maximal class of soft binary constraints in artificial intelligence for which finding the optimal solution can be done in polynomial time, including many common soft and crisp constraints.

Contribution

The paper introduces a maximal tractable class of soft binary constraints, expanding understanding of computational complexity in soft constraint satisfaction problems.

Findings

A polynomial time algorithm exists for the identified class.

The class includes many common soft constraints.

Adding constraints outside this class leads to NP-hard problems.

Abstract

Many researchers in artificial intelligence are beginning to explore the use of soft constraints to express a set of (possibly conflicting) problem requirements. A soft constraint is a function defined on a collection of variables which associates some measure of desirability with each possible combination of values for those variables. However, the crucial question of the computational complexity of finding the optimal solution to a collection of soft constraints has so far received very little attention. In this paper we identify a class of soft binary constraints for which the problem of finding the optimal solution is tractable. In other words, we show that for any given set of such constraints, there exists a polynomial time algorithm to determine the assignment having the best overall combined measure of desirability. This tractable class includes many commonly-occurring soft constraints, such as 'as near as possible' or 'as soon as possible after', as well as crisp constraints such as 'greater than'. Finally, we show that this tractable class is maximal, in the sense that adding any other form of soft binary constraint which is not in the class gives rise to a class of problems which is NP-hard.