Barbara M. Smith

Constraint satisfaction problems: algorithms and applications

A constraint satisfaction problem requires a value, selected from a given finite domain, to be assigned to each variable in the problem, so that all constraints relating the variables are satisfied. Many combinatorial problems in operational research, such as schedulling and timetabling, can be formulated as constraint satisfaction problems. Researchers in artificial intelligence usually adopt a constaint satisfaction approach as their prefererd method when tackling such problems. However constraint satisfaction approches are not widely known amongst operational researchers. The aim of this paper is to introduce constraint statisfaction to the operational researchers.

LOCATING THE PHASE TRANSITION IN BINARY CONSTRAINT SATISFACTION PROBLEMS

Abstract The phase transition in binary constraint satisfaction problems, i.e. the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solutions, as the constraints become tighter, is investigated by examining the behaviour of samples of randomly-generated problems. In contrast to theoretical work, which is concerned with the asymptotic behaviour of problems as the number of variables becomes larger, this paper is concerned with the location of the phase transition in finite problems. The accuracy of a prediction based on the expected number of solutions is discussed; it is shown that the variance of the number of solutions can be used to set bounds on the phase transition and to indicate the accuracy of the prediction. A class of sparse problems, for which the prediction is known to be inaccurate, is considered in detail; it is shown that, for these problems, the phase transition depends on the topology of the constraint graph as well as on the tightness of the constraints.

Random Constraint Satisfaction: Flaws and Structure

A recent theoretical result by Achlioptas et al. shows that many models of random binary constraint satisfaction problems become trivially insoluble as problem size increases. This insolubility is partly due to the presence of ‘flawed variables,’ variables whose values are all ‘flawed’ (or unsupported). In this paper, we analyse how seriously existing work has been affected. We survey the literature to identify experimental studies that use models and parameters that may have been affected by flaws. We then estimate theoretically and measure experimentally the size at which flawed variables can be expected to occur. To eliminate flawed values and variables in the models currently used, we introduce a ‘flawless’ generator which puts a limited amount of structure into the conflict matrix. We prove that such flawless problems are not trivially insoluble for constraint tightnesses up to 1/2. We also prove that the standard models B and C do not suffer from flaws when the constraint tightness is less than the reciprocal of domain size. We consider introducing types of structure into the constraint graph which are rare in random graphs and present experimental results with such structured graphs.

The Progressive Party Problem: Integer Linear Programming and Constraint Programming Compared

Many discrete optimization problems can be formulated as either integer linear programming problems or constraint satisfaction problems. Although ILP methods appear to be more powerful, sometimes constraint programming can solve these problems more quickly. This paper describes a problem in which the difference in performance between the two approaches was particularly marked, since a solution could not be found using ILP.The problem arose in the context of organizing a “progressive party” at a yachting rally. Some yachts were to be designated hosts; the crews of the remaining yachts would then visit the hosts for six successive half-hour periods. A guest crew could not revisit the same host, and two guest crews could not meet more than once. Additional constraints were imposed by the capacities of the host yachts and the crew sizes of the guests.Integer linear programming formulations which included all the constraints resulted in very large models, and despite trying several different strategies, all attempts to find a solution failed. Constraint programming was tried instead and solved the problem very quickly, with a little manual assistance. Reasons for the success of constraint programming in this problem are identified and discussed.

An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem

The constraint satisfaction community has developed a number of heuristics for variable ordering during backtracking search. For example, in conjunction with algorithms which check forwards, the Fail-First (FF) and Brelaz (Bz) heuristics are cheap to evaluate and are generally considered to be very effective. Recent work to understand phase transitions in NP-complete problem classes enables us to compare such heuristics over a large range of different kinds of problems. Furthermore, we are now able to start to understand the reasons for the success, and therefore also the failure, of heuristics, and to introduce new heuristics which achieve the successes and avoid the failures. In this paper, we present a comparison of the Bz and FF heuristics in forward checking algorithms applied to randomly-generated binary CSPs. We also introduce new and very general heuristics and present an extensive study of these. These new heuristics are usually as good as or better than Bz and FF, and we identify problem classes where our new heuristics can be orders of magnitude better. The result is a deeper understanding of what helps heuristics to succeed or fail on hard random problems in the context of forward checking, and the identification of promising new heuristics worthy of further investigation. This research was supported by HCM personal fellowship to the last author, by a University of Strathclyde starter grant to the first author, and by an EPSRC ROPA award GR/K/65706 for the first three authors. Authors listed alphabetically. We thank the other members of the APES group, and our reviewers, for their comments.

Symmetry definitions for constraint satisfaction problems

We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.

Constraint Models for the Covering Test Problem

Covering arrays can be applied to the testing of software, hardware and advanced materials, and to the effects of hormone interaction on gene expression. In this paper we develop constraint programming models of the problem of finding an optimal covering array. Our models exploit global constraints, multiple viewpoints and symmetry-breaking constraints. We show that compound variables, representing tuples of variables in our original model, allow the constraints of this problem to be represented more easily and hence propagate better. With our best integrated model, we are able to either prove the optimality of existing bounds or find new optimal solutions, for arrays of moderate size. Local search on a SAT-encoding of the model is able to find improved solutions and bounds for larger problems.

A BUS CREW SCHEDULING SYSTEM USING A SET COVERING FORMULATION

Abstract A bus crew scheduling system which uses mathematical programming is described. The system is based on a set covering formulation, and includes a number of heuristics to keep the problem to a manageable size. It has been in regular use by London Buses Ltd. since the beginning of 1985 and has been adopted by other bus companies. The crew scheduling problem is described, the solution process is presented and results are discussed briefly.

Random Constraint Satisfaction: Theory Meets Practice

We study the experimental consequences of a recent theoretical result by Achlioptas et al. that shows that conventional models of random problems are trivially insoluble in the limit. We survey the literature to identify experimental studies that lie within the scope of this result. We then estimate theoretically and measure experimentally the size at which problems start to become trivially insoluble. Our results demonstrate that most (but not all) of these experimental studies are luckily unaffected by this result. We also study an alternative model of random problems that does not suffer from this asymptotic weakness. We show that, at a typical problem size used in experimental studies, this model looks similar to conventional models. Finally, we generalize this model so that we can independently adjust the constraint tightness and density.

Symmetry breaking in Graceful Graphs

Symmetry occurs frequently in Constraint Satisfaction Problems (CSPs). For instance, in 3-colouring the nodes of a graph, a CSP model that assigns a specific colour to each node has sets of equivalent solutions in which the three colours are permuted. Symmetry in CSPs can cause wasted search, because the search for solutions may repeatedly visit partial assignments symmetric to ones already considered. If a partial assignment does not lead to a solution, neither will any symmetrically equivalent assignment. When searching for all solutions, for every solution found, all the symmetrically equivalent solutions will also be found.