Abstract. Traditionally, constraint satisfaction has been applied in closedworld scenarios, where all choices and constraints are known from the beginning and fixed. With the Internet, many of the traditional CSP applications in resource allocation, scheduling and planning pose themselves in open-world settings, where choices and constraints are to be discovered from different servers in a network. We examine how such a distributed setting affects changes the assumptions underlying most CSP algorithms, and show how solvers can be augmented with an information gathering component that allows openworld constraint satisfaction. We report on experiments that show strong performance of such methods over others where gathering information and solving the CSP are separated.

. The Golomb ruler problem has been proposed as a challenging constraint satisfaction problem. We consider a large number of different models of this problem, both binary and non-binary. The problem can be modelled using quaternary constraints, but in practice using a set of auxiliary variables and ternary constraints gives better results. A binary encoding of the problem gives a smaller search tree, but is impractical because it takes far longer to run. We compare variable ordering heuristics and consider the use of implied constraints to improve propagation. We believe that more case studies such as this are essential to reduce the skill currently required for successful modelling. 1 Introduction In his AAAI-98 invited talk, Gene Freuder identified modelling as one of the major hurdles preventing the uptake of constraint satisfaction technology. The availability of non-binary constraints can increase the number of possible models of a problem amnd so makes modelling still more diffi...

Fellowship program. 1 There are two well known transformations from non-binary constraints to binary constraints applicable to constraint satisfaction problems (CSPs) with finite domains: the dual transformation and the hidden (variable) transformation. We perform a detailed formal comparison of these two transformations. Our comparison focuses on two backtracking algorithms that maintain a local consistency property at each node in their search tree: the forward checking and maintaining arc consistency algorithms. We first compare local consistency techniques such as arc consistency in terms of their inferential power when they are applied to the original (non-binary) formulation and to each of its binary transformations. For example, we prove that enforcing arc consistency on the original formulation is equivalent to enforcing it on the hidden transformation. We then extend these results to the two backtracking algorithms. We are able to give either a theoretical bound on how much one formulation is better than another, or examples that show such a bound does not exist. For example, we prove that the performance of the forward checking algorithm applied to the hidden transformation of a problem is within a polynomial bound of the performance of the same algorithm applied to the dual transformation of the problem. Our results can be used to help decide if applying one of these transformations to all (or part) of a constraint satisfaction model would be beneficial. 2 1

Abstract: Despite successful application of constraint programming (CP) to solving many real-life problems there is still an indispensable group or researchers considering (wrongly) CP as a simple evaluation technique only. Even if sophisticated search algorithms play an important role in solving constraint-based models, the real power engine behind CP is called constraint propagation (domain filtering, pruning or consistency techniques). In the paper we give a survey of common consistency techniques for binary constraints. We describe the main ideas behind them, list their advantages and limitations, and compare their pruning power. Then we briefly explain how these techniques can be extended to non-binary constraints. Last part of the paper is devoted to modelling issues. We give some hints how the constraint propagation can be exploited more when solving real-life problems. This part is based on our experience with solving real-life programs and it is also supported by empirical observations of other researchers.

We perform a comprehensive study of mappings between constraint satisfaction problems (CSPs) and propositional satisfiability (SAT). We analyse four different mappings of SAT problems into CSPs, and two of CSPs into SAT problems. For each mapping, we compare the impact of achieving arc-consistency on the CSP with unit propagation on the SAT problem. We then extend these results to CSP algorithms that maintain (some level of) arc-consistency during search like FC and MAC, and to the Davis-Putnam procedure (which performs unit propagation at each search node). Because of differences in the branching structure of their search, a result showing the dominance of achieving arc-consistency on the CSP over unit propagation on the SAT problem does not necessarily translate to the dominance of MAC over the DavisPutnam procedure. These results provide insight into the relationship between propositional satisfiability and constraint satisfaction.

Since the origins of the constraint satisfaction paradigm, its restriction to binary constraints has concentrated a significant part of the work. This is understandable because new ideas/techniques are usually much simpler to present/elaborate by first restricting them to the binary case. (See for example the arc consistency algorithms, such as AC-3 or AC-4, which have been presented first in their binary version [10, 12], before being extended to non-binary constraints [11, 13].) But this inclination has highly increased in the early nineties. Authors indeed justied this restriction by the fact that any non-binary constraint network can polyniomally be converted into an equivalent binary one [6, 8, 5, 19]. And, in most cases, they never extended their work to non-binary constraints. Up to now, constraint reasoning has generated robust formal definitions (local consistencies, etc.), original resolution methods (filtering, look-back schemes, decomposition techniques, heuristics, etc.), and theo...

Conway's game of Life provides interesting problems in which modelling issues in constraint programming can be explored. The problem of finding a maximum density stable pattern (`still-life') is discussed. A formulation of this problem as a constraint satisfaction problem with 0-1 variables and non-binary constraints is compared with its dual graph translation into a binary CSP. The success of the dual translation is surprising, from previously-reported experience, since it has as many variables as the non-binary CSP and very large domains. An important factor is the identification of many redundant constraints: it is shown that these can safely be removed from a dual graph translation if arc consistency is maintained during search. 1

Abstract. Constraint satisfaction has been applied with great success in closed-world scenarios, where all options and constraints are known from the beginning and fixed. With the internet, many of the traditional CSP applications in resource allocation, scheduling and planning pose themselves in open-world settings, where options and constraints must be gathered from different agents in a network. We define open constraint optimization as a model of such tasks. Under the assumption that options are discovered in decreasing order of preference, it becomes possible to guarantee optimality even when domains and constraints are not completely known. We propose several algorithms for solving open constraint optimization problems by incrementally gathering options through the network. We report empirical results on their performance on random problems, and analyze how to achieve optimality with a minimal number of queries to the information sources.

In non-binary constraint satisfaction problems, the study of local consistencies that only prune values from domains has so far been largely limited to generalized arc consistency or weaker local consistency properties. This is in contrast with binary constraints where numerous such domain filtering consistencies have been proposed. In this paper we present a detailed theoretical, algorithmic and empirical study of domain filtering consistencies for non-binary problems. We study three domain filtering consistencies that are inspired by corresponding variable based domain filtering consistencies for binary problems. These consistencies are stronger than generalized arc consistency, but weaker than pairwise consistency, which is a strong consistency that removes tuples from constraint relations. Among other theoretical results, and contrary to expectations, we prove that these new consistencies do not reduce to the variable based definitions of their counterparts on binary constraints. We propose a number of algorithms to achieve the three consistencies. One of these algorithms has a time complexity comparable to that for generalized arc consistency despite performing more pruning. Experiments demonstrate that our new consistencies are promising as they can be more efficient than generalized arc consistency on certain non-binary problems.

One of the challenges for the constraint satisfaction community has been to develop an automated approach to solving Constraint Satisfaction Problems (CSPs) rather than creating specific algorithms for specific problems. Much of this work has concentrated on the development and improvement of general purpose backtracking techniques. However, the success of relatively simple local search techniques on larger satisfiability problems [Selman et al. 1992] and CSPs such as the n-queens [Minton et al. 1992] has caused interest in applying local search to constraint satisfaction. In this thesis we look at the usefulness of constraint weighting as a local search technique for constraint satisfaction. The work is based on the clause weighting ideas of Selman and Kautz [1993] and Morris [1993] and applies, evaluates and extends these ideas from the satisfiability domain to the more general domain of CSPs. Specifically, the contributions of the thesis are: The introduction of a local search taxonomy. We examine the various better known local search techniques and recognise four basic strategies: restart, randomness, memory and weighting.