Improved Algorithms for Allen's Interval Algebra by Dynamic Programming with Sublinear Partitioning

• URL: http://arxiv.org/abs/2305.15950v1
• Date: Thu, 25 May 2023 11:45:12 GMT
• Title: Improved Algorithms for Allen's Interval Algebra by Dynamic Programming with Sublinear Partitioning
• Authors: Leif Eriksson and Victor Lagerkvist
• Abstract summary: Allen's interval algebra is one of the most well-known calculi in qualitative temporal reasoning. We propose a novel framework for solving NP-hard qualitative reasoning problems. We obtain a major improvement of $O*((fraccnlogn)n)$ for Allen's interval algebra.
• Score: 9.594432031144716
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Allen's interval algebra is one of the most well-known calculi in qualitative temporal reasoning with numerous applications in artificial intelligence. Recently, there has been a surge of improvements in the fine-grained complexity of NP-hard reasoning tasks, improving the running time from the naive $2^{O(n^2)}$ to $O^*((1.0615n)^{n})$, with even faster algorithms for unit intervals a bounded number of overlapping intervals (the $O^*(\cdot)$ notation suppresses polynomial factors). Despite these improvements the best known lower bound is still only $2^{o(n)}$ (under the exponential-time hypothesis) and major improvements in either direction seemingly require fundamental advances in computational complexity. In this paper we propose a novel framework for solving NP-hard qualitative reasoning problems which we refer to as dynamic programming with sublinear partitioning. Using this technique we obtain a major improvement of $O^*((\frac{cn}{\log{n}})^{n})$ for Allen's interval algebra. To demonstrate that the technique is applicable to more domains we apply it to a problem in qualitative spatial reasoning, the cardinal direction point algebra, and solve it in $O^*((\frac{cn}{\log{n}})^{2n/3})$ time. Hence, not only do we significantly advance the state-of-the-art for NP-hard qualitative reasoning problems, but obtain a novel algorithmic technique that is likely applicable to many problems where $2^{O(n)}$ time algorithms are unlikely.

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