Steven David Prestwich

Synthesizing filtering algorithms for global chance-constraints

Stochastic Constraint Satisfaction Problems (SCSPs) are a powerful modeling framework for problems under uncertainty. To solve them is a P-Space task. The only solution approach to date compiles down SCSPs into classical CSPs. This allows the reuse of classical constraint solvers to solve SCSPs, but at the cost of increased space requirements and weak constraint propagation. This paper tries to overcome some of these drawbacks by automatically synthesizing filtering algorithms for global chance-constraints. These filtering algorithms are parameterized by propagators for the deterministic version of the chance-constraints. This approach allows the reuse of existing propagators in current constraint solvers and it enhances constraint propagation. Experiments show the benefits of this novel approach.

Cost-based filtering for stochastic inventory control

An interesting class of production/inventory control problems considers a single product and a single stocking location, given a stochastic demand with a known non-stationary probability distribution. Under a widely-used control policy for this type of inventory system, the objective is to find the optimal number of replenishments, their timings and their respective order-up-to-levels that meet customer demands to a required service level. We extend a known CP approach for this problem using a cost-based filtering method. Our algorithm can solve to optimality instances of realistic size much more efficiently than previous approaches, often with no search effort at all.

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 Hybrid Seachr Architecture Applied to Hard Random 3-SAT and Low-Autocorrelation Binary Sequences

The hybridisation of systematic and stochastic search is an active research area with potential benefits for real-world combinatorial problems. This paper shows that randomising the backtracking component of a systematic backtracker can improve its scalability to equal that of stochastic local search. The hybrid may be viewed as stochastic local search in a constrained space, cleanly combininglo cal search with constraint programming techniques. The approach is applied to two very different problems. Firstly a hybrid of local search and constraint propagation is applied to hard random 3-SAT problems, and is the first constructive search algorithm to solve very large instances. Secondly a hybrid of local search and branch-and-bound is applied to low-autocorrelation binary sequences (a notoriously difficult communications engineering problem), and is the first stochastic search algorithm to find optimal solutions. These results show that the approach is a promising one for both constraint satisfaction and optimisation problems.

Random walk with continuously smoothed variable weights

Many current local search algorithms for SAT fall into one of two classes. Random walk algorithms such as Walksat/SKC, Novelty+ and HWSAT are very successful but can be trapped for long periods in deep local minima. Clause weighting algorithms such as DLM, GLS, ESG and SAPS are good at escaping local minima but require expensive smoothing phases in which all weights are updated. We show that Walksat performance can be greatly enhanced by weighting variables instead of clauses, giving the best known results on some benchmarks. The new algorithm uses an efficient weight smoothing technique with no smoothing phase.

Combining the Scalability of Local Search with the Pruning Techniques of Systematic Search

Systematic backtracking is used in many constraint solvers and combinatorial optimisation algorithms. It is complete and can be combined with powerful search pruning techniques such as branch-and-bound, constraint propagation and dynamic variable ordering. However, it often scales poorly to large problems. Local search is incomplete, and has the additional drawback that it cannot exploit pruning techniques, making it uncompetitive on some problems. Nevertheless its scalability makes it superior for many large applications. This paper describes a hybrid approach called Incomplete Dynamic Backtracking, a very flexible form of backtracking that sacrifices completeness to achieve the scalability of local search. It is combined with forward checking and dynamic variable ordering, and evaluated on three combinatorial problems: on the n-queens problem it out-performs the best local search algorithms; it finds large optimal Golomb rulers much more quickly than a constraint-based backtracker, and better rulers than a genetic algorithm; and on benchmark graphs it finds larger cliques than almost all other tested algorithms. We argue that this form of backtracking is actually local search in a space of consistent partial assignments, offering a generic way of combining standard pruning techniques with local search.

Hard and soft constraints for reasoning about qualitative conditional preferences

Many real life optimization problems are defined in terms of both hard and soft constraints, and qualitative conditional preferences. However, there is as yet no single framework for combined reasoning about these three kinds of information. In this paper we study how to exploit classical and soft constraint solvers for handling qualitative preference statements such as those captured by the CP-nets model. In particular, we show how hard constraints are sufficient to model the optimal outcomes of a possibly cyclic CP-net, and how soft constraints can faithfully approximate the semantics of acyclic conditional preference statements whilst improving the computational efficiency of reasoning about these statements.

Negative Effects of Modeling Techniques on Search Performance

Symmetries occur in many combinatorial problems, and a great deal of research has been done on symmetry breaking techniques for backtrack search. However, few results have been reported on the use of symmetry breaking with local search. On four classes of problem we find that adding symmetry breaking constraints to a model impairs local search performance, in terms of both execution time and search steps. We also find that implied constraints can impair backtrack search performance. These results show that modeling techniques and search heuristics should be combined with caution. They also motivate a novel modeling technique for local search: removing constraints to add new symmetries.

Local Search on SAT-encoded Colouring Problems

Constraint satisfaction problems can be SAT-encoded in more than one way, and the choice of encoding can be as important as the choice of search algorithm. Theoretical results are few but experimental comparisons have been made between encodings, using both local and backtrack search algorithms. This paper compares local search performance on seven encodings of graph colouring benchmarks. Two of the encodings are new and one of them gives generally better results than known encodings. We also find better results than expected for two variants of the log encoding, and surprisingly poor results for the support encoding.

A Global Chance-Constraint for Stochastic Inventory Systems Under Service Level Constraints

We consider a class of production/inventory control problems that has a single product and a single stocking location, for which a stochastic demand with a known non-stationary probability distribution is given. Under the widely-known replenishment cycle policy the problem of computing policy parameters under service level constraints has been modeled using various techniques. Tarim and Kingsman introduced a modeling strategy that constitutes the state-of-the-art approach for solving this problem. In this paper we identify two sources of approximation in Tarim and Kingsman’s model and we propose an exact stochastic constraint programming approach. We build our approach on a novel concept, global chance-constraints, which we introduce in this paper. Solutions provided by our exact approach are employed to analyze the accuracy of the model developed by Tarim and Kingsman.