Abstract

Many computational problems arising in, for instance, artificial intelligence can be realized as infinite-domain constraint satisfaction problems (CSPs) based on partition schemes: a set of pairwise disjoint binary relations (containing the equality relation) whose union spans the underlying domain and which is closed under converse. We first consider partition schemes that contain an acyclic order and where the constraint language contains all unions of the basic relations; such CSPs are frequently occurring in e.g. temporal and spatial reasoning. We identify properties of such orders which, when combined, are sufficient to establish NP-hardness of the CSP and strong lower bounds under the exponential-time hypothesis, even for degree-bounded problems. This result explains, in a uniform way, many existing hardness results from the literature, and shows that it is impossible to obtain subexponential time algorithms unless the exponential-time hypothesis fails. However, some of these problems (including several important temporal problems), despite likely not being solvable in subexponential time, admit non-trivial improved exponential-time algorithm, and we present a novel improved algorithm for RCC-8 and related formalisms.

Highlights

  • In this article we study the complexity of infinite-domain constraint satisfaction problems over partition schemes

  • One way of interpreting these results is that constraint satisfaction problems (CSPs)(B∨=), when B is a partition scheme containing an acyclic order satisfying the aforementioned properties, is far from being polynomial-time solvable: there is a constant c > 1 such that the problem cannot be solved in O time

  • We have established that CSP(B∨=) for B ∈ H is unlikely to be solvable in subexponential time, so we focus our efforts on constructing faster exponential-time algorithms

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Summary

Introduction

In this article we study the complexity of infinite-domain constraint satisfaction problems over partition schemes. In this framework one can formulate many naturally occurring reasoning problems in artificial intelligence such as Allen’s interval algebra and the region connection calculus. To the best of our knowledge, this is the first lower bound under the exponentialtime hypothesis for problems of this form. Motivated by these lower bounds we turn to the problem of constructing improved algorithms for CSPs over partition schemes, with a particular focus on the region connection calculus

Background
Our results
Preliminaries
Acyclic orders
Conditions on acyclic orders
Examples
ETH-based lower bounds and NP-hardness
Consequences
A tractable subclass of degree-bounded problems
The branching algorithm
A faster algorithm for RCC-8
Discussion
Full Text
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