Peter Jonsson

Reasoning about temporal relations: The tractable subalgebras of Allen's interval algebra

Allens interval algebra is one of the best established formalisms for temporal reasoning. This article provides the final step in the classification of complexity for satisfiability problems over constraints expressed in this algebra. When the constraints are chosen from the full Allens algebra, this form of satisfiability problem is known to be NP-complete. However, eighteen tractable subalgebras have previously been identified; we show here that these subalgebras include all possible tractable subsets of Allens algebra. In other words, we show that this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NP-complete. We obtain this dichotomy result by giving a new uniform description of the known maximal tractable subalgebras, and then systematically using a general algebraic technique for identifying maximal subalgebras with a given property.

A unifying approach to temporal constraint reasoning

Abstract We present a formalism, Disjunctive Linear Relations (DLRs), for reasoning about temporal constraints. DLRs subsume most of the formalisms for temporal constraint reasoning proposed in the literature and is therefore computationally expensive. We also present a restricted type of DLRs, Horn DLRs, which have a polynomial-time satisfiability problem. We prove that most approaches to tractable temporal constraint reasoning can be encoded as Horn DLRs, including the ORD-Horn algebra by Nebel and Burckert and the simple temporal constraints by Dechter et al. Thus, DLRs is a suitable unifying formalism for reasoning about temporal constraints.

State-variable planning under structural restrictions: algorithms and complexity

Abstract Computationally tractable planning problems reported in the literature so far have almost exclusively been defined by syntactical restrictions. To better exploit the inherent structure in problems, it is probably necessary to study also structural restrictions on the underlying state-transition graph. The exponential size of this graph, though, makes such restrictions costly to test. Hence, we propose an intermediate approach, using a state-variable model for planning and defining restrictions on the separate state-transition graphs for each state variable. We identify such restrictions which can tractably be tested and we present a planning algorithm which is correct and runs in polynomial time under these restrictions. The algorithm has been implemented and it outperforms Graphplan on a number of test instances. In addition, we present an exhaustive map of the complexity results for planning under all combinations of four previously studied syntactical restrictions and our five new structural restrictions. This complexity map considers both the optimal and non-optimal plan generation problem.

Counting models for 2 SAT and 3 SAT formulae

We here present algorithms for counting models and max-weight models for 2SAT and 3SAT formulae. They use polynomial space and run in O(1.2561n) and O(1.6737n) time, respectively, where n is the number of variables. This is faster than the previously best algorithms for counting nonweighted models for 2SAT and 3SAT, which run in O(1.3247n) and O(1.6894n) time, respectively. In order to prove these time bounds, we develop new measures of formula complexity, allowing us to conveniently analyze the effects of certain factors with a large impact on the total running time. We also provide an algorithm for the restricted case of separable 2SAT formulae, with fast running times for well-studied input classes. For all three algorithms we present interesting applications, such as computing the permanent of sparse 0/1 matrices.

Some Results on the Complexity of Planning with Incomplete Information

Planning with incomplete information may mean a number of different things; that certain facts of the initial state are not known, that operators can have random or nondeterministic effects, or that the plans created contain sensing operations and are branching. Study of the complexity of incomplete information planning has so far been concentrated on probabilistic domains, where a number of results have been found. We examine the complexity of planning in nondeterministic propositional domains. This differs from domains involving randomness, which has been well studied, in that for a nondeterministic choice, not even a probability distribution over the possible outcomes is known. The main result of this paper is that the non-branching plan existence problem in unobservable domains with an expressive operator formalism is EXPSPACE-complete. We also discuss several restrictions, which bring the complexity of the problem down to PSPACE-complete, and extensions to the fully and partially observable cases.

The complexity of counting homomorphisms seen from the other side

For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B. Grohe has proved that, under a certain complexity-theoretic assumption, HOM(C, _) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width. We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C, _) is solvable in polynomial time if and only if all structures in C have bounded tree-width. This answers an open question posed by Grohe.

A complete classification of tractability in RCC-5

We investigate the computational properties of the spatial algebra RCC-5 which is a restricted version of the RCC framework for spatial reasoning. The satisfiability problem for RCC-5 is known to be NP-complete but not much is known about its approximately four billion subclasses. We provide a complete classification of satisfiability for all these subclasses into polynomial and NP-complete respectively. In the process, we identify all maximal tractable subalgebras which are four in total.

Essential convexity and complexity of semi-algebraic constraints

Let \Gamma be a structure with a finite relational signature and a first-order definition in (R;*,+) with parameters from R, that is, a relational structure over the real numbers where all relation ...

The Approximability of Three-valued MAX CSP

In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints. This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem. It is known that every Boolean (that is, two-valued) MAX CSP with a finite set of allowed constraint types is either solvable exactly in polynomial time or else APX-complete (and hence can have no polynomial-time approximation scheme unless P=NP). It has been an open problem for several years whether this result can be extended to non-Boolean MAX CSP, which is much more difficult to analyze than the Boolean case. In this paper, we make the first step in this direction by establishing this result for MAX CSP over a three-element domain. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known algebraic combinatorial property of supermodularity. We also show that every hard three-valued MAX CSP contains, in a certain specified sense, one of the two basic hard MAX CSPs which are the Maximum k-Colorable Subgraph problems for k=2,3.

Eight maximal tractable subclasses of Allen's algebra with metric time

This paper combines two important directions of research in temporal resoning: that of finding maximal tractable subclasses of Allens interval algebra, and that of reasoning with metric temporal information. Eight new maximal tractable subclasses of Allens interval algebra are presented, some of them subsuming previously reported tractable algebras. The algebras allow for metric temporal constraints on interval starting or ending points, using the recent framework of Horn DLRs. Two of the algebras can express the notion of sequentiality between intervals, being the first such algebras admitting both qualitative and metric time.