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

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 $IA$.