A Multivariate Complexity Analysis of Qualitative Reasoning Problems

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

An initial study of time complexity in infinite-domain constraint satisfaction

The constraint satisfaction problem (CSP) is a widely studied problem with numerous applications in computer science and artificial intelligence. For infinite-domain CSPs, there are many results separating tractable and NP-hard cases while upper and lower bounds on the time complexity of hard cases are virtually unexplored. Hence, we initiate a study of the worst-case time complexity of such CSPs. We analyze backtracking algorithms and determine upper bounds on their time complexity. We present asymptotically faster algorithms based on enumeration techniques and we show that these algorithms are applicable to well-studied problems in, for instance, temporal reasoning. Finally, we prove non-trivial lower bounds applicable to many interesting CSPs, under the assumption that certain complexity-theoretic assumptions hold. The gap between upper and lower bounds is in many cases surprisingly small, which suggests that our upper bounds cannot be significantly improved

An initial study of time complexity in infinite-domain constraint satisfaction

The constraint satisfaction problem (CSP) is a widely studied problem with numerous applications in computer science and artificial intelligence. For infinite-domain CSPs, there are many results separating tractable and NP-hard cases while upper and lower bounds on the time complexity of hard cases are virtually unexplored, Hence, we initiate a study of the worst-case time complexity of such CSPs, We analyze backtracking algorithms and determine upper bounds on their time complexity. We present asymptotically faster algorithms based on enumeration techniques and we show that these algorithms are applicable to well-studied problems in, for instance, temporal reasoning. Finally, we prove non-trivial lower bounds applicable to many interesting CSPs, under the assumption that certain complexity-theoretic assumptions hold. The gap between upper and lower bounds is in many cases surprisingly small, which suggests that our upper bounds cannot be significantly improved. (C) 2017 Elsevier B.V. All rights reserved.Funding Agencies|Swedish Research Council [621-2012-3239]; National Graduate School in Computer Science (CUGS), Sweden; DFG project "Homogene Strukturen, Bedingungserfullungsprobleme, and topologische Klone" [622397]</p