Pure Nash Equilibria: Hard and Easy Games

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems

A Backtracking-Based Algorithm for Computing Hypertree-Decompositions

Hypertree decompositions of hypergraphs are a generalization of tree decompositions of graphs. The corresponding hypertree-width is a measure for the cyclicity and therefore tractability of the encoded computation problem. Many NP-hard decision and computation problems are known to be tractable on instances whose structure corresponds to hypergraphs of bounded hypertree-width. Intuitively, the smaller the hypertree-width, the faster the computation problem can be solved. In this paper, we present the new backtracking-based algorithm det-k-decomp for computing hypertree decompositions of small width. Our benchmark evaluations have shown that det-k-decomp significantly outperforms opt-k-decomp, the only exact hypertree decomposition algorithm so far. Even compared to the best heuristic algorithm, we obtained competitive results as long as the hypergraphs are not too large.Comment: 19 pages, 6 figures, 3 table

On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions

The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a computation problem where the focus is either on finding one solution, or on enumerating all solutions, possibly projected to some given set of output variables. The paper investigates the structural tractability of the problem of enumerating (possibly projected) solutions, where tractability means here computable with polynomial delay (WPD), since in general exponentially many solutions may be computed. A general framework based on the notion of tree projection of hypergraphs is considered, which generalizes all known decomposition methods. Tractability results have been obtained both for classes of structures where output variables are part of their specification, and for classes of structures where computability WPD must be ensured for any possible set of output variables. These results are shown to be tight, by exhibiting dichotomies for classes of structures having bounded arity and where the tree decomposition method is considered

Towards Efficient Reasoning under Guarded-based Disjunctive Existential Rules

International audienceThe complete picture of the complexity of answering (unions of) conjunctive queries under the main guarded-based classes of disjunc- tive existential rules has been recently settled. It has been shown that the problem is very hard, namely 2ExpTime-complete, even for fixed sets of rules expressed in lightweight formalisms. This gives rise to the question whether its complexity can be reduced by restricting the query language. Several subclasses of conjunctive queries have been proposed with the aim of reducing the complexity of classical database problems such as query evaluation and query containment. Three of the most prominent subclasses of this kind are queries of bounded hypertree-width, queries of bounded treewidth and acyclic queries. The central objective of the present paper is to understand whether the above query languages have a positive impact on the complexity of query answering under the main guarded-based classes of disjunctive existential rules. We show that (unions of) conjunctive queries of bounded hypertree- width and of bounded treewidth do not reduce the complexity of our problem, even if we focus on predicates of bounded arity, or on fixed sets of disjunctive existential rules. Regarding acyclic queries, although our problem remains 2ExpTime-complete in general, in some relevant set- tings the complexity reduces to ExpTime-complete; in fact, this requires to bound the arity of the predicates, and for some expressive guarded- based formalisms, to fix the set of rules

High precision Monte Carlo study of the 3D XY-universality class

We present a Monte Carlo study of the two-component $\phi^4$ model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant $\lambda$ we eliminate leading order corrections to scaling. High statistics simulations using finite size scaling techniques yield $\nu=0.6723(3)[8]$ and $\eta=0.0381(2)[2]$, where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.Comment: 13 page

The Complexity of Reasoning for Fragments of Default Logic

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as \SigmaPtwo-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.Comment: Corrected versio

Distributed XML Design

A distributed XML document is an XML document that spans several machines. We assume that a distribution design of the document tree is given, consisting of an XML kernel-document T[f1,...,fn] where some leaves are "docking points" for external resources providing XML subtrees (f1,...,fn, standing, e.g., for Web services or peers at remote locations). The top-down design problem consists in, given a type (a schema document that may vary from a DTD to a tree automaton) for the distributed document, "propagating" locally this type into a collection of types, that we call typing, while preserving desirable properties. We also consider the bottom-up design which consists in, given a type for each external resource, exhibiting a global type that is enforced by the local types, again with natural desirable properties. In the article, we lay out the fundamentals of a theory of distributed XML design, analyze problems concerning typing issues in this setting, and study their complexity.Comment: "56 pages, 4 figures

Dyadic existential rules

In the field of ontology-based query answering, existential rules (a.k.a. tuple-generating dependencies) form an expressive Datalog-based language to specify implicit knowledge. The presence of existential quantification in rule-heads, however, makes the main reasoning tasks undecidable. To overcome this limitation, in the last two decades, a number of classes of existential rules guaranteeing the decidability of query answering have been proposed. Unfortunately, such classes are typically based on different syntactic conditions imposing the development of different ad hoc reasoners. This paper introduces a novel general condition that allows to define, systematically, from any decidable class C of existential rules, a new class called Dyadic-C that enjoys the following properties: (i) it is decidable; (ii) it generalizes C; (iii) it keeps the same data complexity as C; and (iv) it can exploit any reasoner for query answering over C. Additionally, the paper proposes a simple and elegant syntactic condition that gives rise to the class Ward+ generalizing the well-known decidable classes Shy and Ward, and being included in Dyadic-Shy

The XY Model and the Three-state Antiferromagnetic Potts model in Three Dimensions: Critical Properties from Fluctuating Boundary Conditions

We present the results of a Monte Carlo study of the three-dimensional XY model and the three-dimensional antiferromagnetic three-state Potts model. In both cases we compute the difference in the free energies of a system with periodic and a system with antiperiodic boundary conditions in the neighbourhood of the critical coupling. From the finite-size scaling behaviour of this quantity we extract values for the critical temperature and the critical exponent nu that are compatible with recent high statistics Monte Carlo studies of the models. The results for the free energy difference at the critical temperature and for the exponent nu confirm that both models belong to the same universality class.Comment: 13 pages, latex-file+2 ps-files KL-TH-94/8 and CERN-TH.7290/9