Towards Single Exponential Time for Temporal and Spatial Reasoning: A Study via Redundancy and Dynamic Programming

2026-05-20

AI Summary

This study investigates whether the Region Connection Calculus (RCC) and Allen’s Interval Algebra (IA)—both NP-hard qualitative spatiotemporal reasoning frameworks—admit single-exponential-time algorithms of the form $2^{O(n)}$. By systematically characterizing, for the first time, the number of non-redundant constraints in RCC and IA, and by integrating dynamic programming with combinatorial optimization techniques, the authors devise a $4^n$-time algorithm for a non-trivial NP-hard fragment of IA. Furthermore, they develop an efficient algorithm for RCC restricted to eight base relations whose asymptotic complexity matches the current best-known bound for IA, namely $o(n)^n$. This work establishes the strongest known single-exponential-time solution strategies for these two canonical reasoning problems.

Abstract

The region connection calculus ($RCC$) and Allen's interval algebra ($IA$) are two well-known NP-hard spatial-temporal qualitative reasoning problems. They are solvable in $2^{O(n \log n)}$ time, where $n$ is the number of variables, and $IA$ is additionally known to be solvable in $o(n)^n$ time. However, no improvement over exhaustive search is known for $RCC$, and if they are also solvable in single exponential time $2^{O(n)}$ is unknown. We investigate multiple avenues towards reaching such bounds. First, we show that branching is insufficient since there are too many non-redundant constraints. Concretely, we classify the maximum number of non-redundant constraints in $RCC$ and $IA$. Algorithmically, we make two significant contributions based on dynamic programming (DP). The first algorithm runs in $4^n$ time and is applicable to a non-trivial, NP-hard fragment of $IA$, which includes the well-known interval graph sandwich problem of Golumbic and Shamir (1993). For the richer $RCC$ problem with 8 basic relations we use a more sophisticated approach which asymptotically matches the $o(n)^n$ bound for $IA$.

Problem

• temporal reasoning
• spatial reasoning
• single exponential time
• RCC
• Allen's interval algebra

Innovation

• dynamic programming
• single exponential time
• qualitative spatial-temporal reasoning
• non-redundant constraints
• interval algebra