A Multivariate Complexity Analysis of Qualitative Reasoning Problems

TL;DR

This paper introduces a multivariate analysis approach to develop single-exponential algorithms for qualitative reasoning problems in AI, focusing on parameters like variables and problem width.

Contribution

It defines new problem classes FPE and XE, and demonstrates their applicability to temporal reasoning and Allen's interval algebra, advancing algorithmic complexity understanding.

Findings

Partially Ordered Time problem solvable in 16^{kn} time

Interval algebra network consistency solvable in (2nk)^{2k} * 2^{n} time

Multivariate analysis can generalize to other qualitative reasoning problems.

Abstract

Qualitative reasoning is an important subfield of artificial intelligence where one describes relationships with qualitative, rather than numerical, relations. Many such reasoning tasks, e.g., Allen's interval algebra, can be solved in $2^{O(n \cdot \log n)}$ time, but single-exponential running times $2^{O(n)}$ are currently far out of reach. In this paper we consider single-exponential algorithms via a multivariate analysis consisting of a fine-grained parameter $n$ (e.g., the number of variables) and a coarse-grained parameter $k$ expected to be relatively small. We introduce the classes FPE and XE of problems solvable in $f(k) \cdot 2^{O(n)}$, respectively $f(k)^n$, time, and prove several fundamental properties of these classes. We proceed by studying temporal reasoning problems and (1) show that the Partially Ordered Time problem of effective width $k$ is solvable in $16^{kn}$ time and is thus included in XE, and (2) that the network consistency problem for Allen's interval algebra with no interval overlapping with more than $k$ others is solvable in $(2nk)^{2k} \cdot 2^{n}$ time and is included in FPE. Our multivariate approach is in no way limited to these to specific problems and may be a generally useful approach for obtaining single-exponential algorithms.