Improved Algorithms for Allen's Interval Algebra: a Dynamic Programming Approach

Abstract

The constraint satisfaction problem (CSP) is an important framework in artificial intelligence used to model e.g. qualitative reasoning problems such as Allen's interval algebra A. There is strong practical incitement to solve CSPs as efficiently as possible, and the classical complexity of temporal CSPs, including A, is well understood. However, the situation is more dire with respect to running time bounds of the form O(f(n)) (where n is the number of variables) where existing results gives a best theoretical upper bound 2^O(n * log n) which leaves a significant gap to the best (conditional) lower bound 2^o(n). In this paper we narrow this gap by presenting two novel algorithms for temporal CSPs based on dynamic programming. The first algorithm solves temporal CSPs limited to constraints of arity three in O(3^n) time, and we use this algorithm to solve A in O((1.5922n)^n) time. The second algorithm tackles A directly and solves it in O((1.0615n)^n), implying a remarkable improvement over existing methods since no previously published algorithm belongs to O((cn)^n) for any c. We also extend the latter algorithm to higher dimensions box algebras where we obtain the first explicit upper bound.