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.

翻译：Allen区间代数是定性时态推理中最著名的演算之一，在人工智能领域具有广泛应用。近期，针对NP困难推理任务的细粒度复杂度研究取得了一系列进展，将运行时间从朴素的$2^{O(n^2)}$提升至$O^*((1.0615n)^{n})$，并且针对有限重叠区间的单位区间问题存在更快速的算法（$O^*(\cdot)$表示忽略多项式因子）。尽管取得这些进展，已知的最佳下界仍仅为$2^{o(n)}$（基于指数时间假设），任何方向上的重大突破似乎都需要计算复杂度的根本性进展。本文提出一种解决NP困难定性推理问题的新框架——动态规划与亚线性划分。利用该技术，我们将Allen区间代数的复杂度显著提升至$O^*((\frac{cn}{\log{n}})^{n})$。为证明该技术的广泛适用性，我们将其应用于定性空间推理中的基点方向代数问题，并实现了$O^*((\frac{cn}{\log{n}})^{2n/3})$的运行时间。因此，我们不仅显著推进了NP困难定性推理问题的最新进展，还获得了一种可能适用于许多$2^{O(n)}$时间算法难以实现的问题的新型算法技术。