We develop an efficient algorithm for computing pure strategy Nash equilibria that satisfy various criteria (such as the utilitarian or Nash-Bernoulli social welfare functions) in games with sparse interaction structure. Our algorithm, called Valued Nash Propagation (VNP), integrates the optimisation problem of maximising a criterion with the constraint satisfaction problem of finding a game's equilibria to construct a criterion that defines a c -semiring. Given a suitably compact game structure, this criterion can be efficiently optimised using message-passing. To this end, we first show that VNP is complete in games whose interaction structure forms a hypertree. Then, we go on to provide theoretic and empirical results justifying its use on games with arbitrary structure; in particular, we show that it computes the optimum >82% of the time and otherwise selects an equilibrium that is always within 2% of the optimum on average.

Dominance testing, the problem of determining whether an outcome is preferred over another, is of fundamental importance in many applications. Hence, there is a need for algorithms and tools for dominance testing. CP-nets and TCP-nets are some of the widely studied languages for representing and reasoning with preferences. We reduce dominance testing in TCP-nets to reachability analysis in a graph of outcomes. We provide an encoding of TCP-nets in the form of a Kripke structure for CTL. We show how to compute dominance using NuSMV, a model checker for CTL. We present results of experiments that demonstrate the feasibility of our approach to dominance testing.

The next challenge in qualitative spatial and temporal reasoning is to develop calculi that deal with different aspects of space and time. One approach to achieve this is to combine existing calculi that cover the different aspects. This, however, can lead to calculi that have a very large number of relations and it is a matter of ongoing discussions within the research community whether such large calculi are too large to be useful. In this paper we develop a procedure for reasoning about some of the largest known calculi, the Rectangle Algebra and the Block Algebra with about 10 661  relations. We demonstrate that reasoning over these calculi is possible and can be done efficiently in many cases. This is a clear indication that one of the main goals of the field can be achieved: highly expressive spatial and temporal representations that support efficient reasoning.

PCS is a CSP solver that can produce a machine-checkable deductive proof in case it decides that the input problem is unsatisfiable. The roots of the proof may be nonclausal constraints, whereas the rest of the proof is based on resolution of signed clauses, ending with the empty clause. PCS uses parameterized, constraint-specific inference rules in order to bridge between the nonclausal and the clausal parts of the proof. The consequent of each such rule is a signed clause that is 1) logically implied by the nonclausal premise, and 2) strong enough to be the premise of the consecutive proof steps. The resolution process itself is integrated in the learning mechanism, and can be seen as a generalization to CSP of a similar solution that is adopted by competitive SAT solvers.

We present a highly efficient incremental algorithm for propagating bounded knapsack constraints. Our algorithm is based on the sublinear filtering algorithm for binary knapsack constraints by Katriel et al. and achieves similar speed-ups of one to two orders of magnitude when compared with its linear-time counterpart. We also show that the representation of bounded knapsacks as binary knapsacks leads to ineffective filtering behavior. Experiments on standard knapsack benchmarks show that the new algorithm significantly outperforms existing methods for handling bounded knapsack constraints.

Weighted Constraint Satisfaction is made practical by powerful consistency techniques, such as AC*, FDAC* and EDAC*, which reduce search space effectively and efficiently during search, but they are designed for only binary and ternary constraints. To allow soft global constraints, usually of high arity, to enjoy the same benefits, Lee and Leung give polynomial time algorithms to enforce generalized AC* (GAC*) and FDAC* (FDGAC*) for projection-safe soft non-binary constraints. Generalizing the stronger EDAC* is less straightforward. In this paper, we first reveal the oscillation problem when enforcing EDAC* on constraints sharing more than one variable. To avoid oscillation, we propose a weak version of EDAC* and generalize it to weak EDGAC* for non-binary constraints. Weak EDGAC* is stronger than FDGAC* and GAC*, but weaker than VAC and soft k -consistency for k > 2. We also show that weak EDGAC* can be enforced in polynomial time for projection-safe constraints. Extensive experimentation confirms the efficiency of our proposal.

Consistency properties and algorithms for achieving them are at the heart of the success of Constraint Programming. In this paper, we study the relational consistency property R ( *,m ) C, which is equivalent to m-wise consistency proposed in relational databases. We also define wR ( *,m ) C, a weaker variant of this property. We propose an algorithm for enforcing these properties on a Constraint Satisfaction Problem by tightening the existing relations and without introducing new ones. We empirically show that wR(*,m)C solves in a backtrack-free manner all the instances of some CSP benchmark classes, thus hinting at the tractability of those classes.

Planning as satisfiability is a principal approach to planning with many eminent advantages. The existing planning as satisfiability techniques usually use encodings compiled from the STRIPS formalism. We introduce a novel SAT encoding scheme based on the SAS+ formalism. It exploits the structural information in the SAS+ formalism, resulting in more compact SAT instances and reducing the number of clauses by up to 50 fold. Our results show that this encoding scheme improves upon the STRIPS-based encoding, in terms of both time and memory efficiency.

We define the concept of an internal symmetry. This is a symmety within a solution of a constraint satisfaction problem. We compare this to solution symmetry, which is a mapping between different solutions of the same problem. We argue that we may be able to exploit both types of symmetry when finding solutions. We illustrate the potential of exploiting internal symmetries on two benchmark domains: Van der Waerden numbers and graceful graphs. By identifying internal symmetries we are able to extend the state of the art in both cases.

We study propagation algorithms for the conjunction of two AllDifferent constraints. Solutions of an AllDifferent constraint can be seen as perfect matchings on the variable/value bipartite graph. Therefore, we investigate the problem of finding simultaneous bipartite matchings. We present an extension of the famous Hall theorem which characterizes when simultaneous bipartite matchings exists. Unfortunately, finding such matchings is NP-hard in general. However, we prove a surprising result that finding a simultaneous matching on a convex bipartite graph takes just polynomial time. Based on this theoretical result, we provide the first polynomial time bound consistency algorithm for the conjunction of two AllDifferent constraints. We identify a pathological problem on which this propagator is exponentially faster compared to existing propagators. Our experiments show that this new propagator can offer significant benefits over existing methods.