Patrick Prosser

HYBRID ALGORITHMS FOR THE CONSTRAINT SATISFACTION PROBLEM

It might be said that there are five basic tree search algorithms for the constraint satisfaction problem (csp), namely, naive backtracking (BT), backjumping (BJ), conflict‐directed backjumping (CBJ), backmarking (BM), and forward checking (FC). In broad terms, BT, BJ, and CBJ describe different styles of backward move (backtracking), whereas BT, BM, and FC describe different styles of forward move (labeling of variables). This paper presents an approach that allows base algorithms to be combined, giving us new hybrids. The base algorithms are described explicitly, in terms of a forward move and a backward move. It is then shown that the forward move of one algorithm may be combined with the backward move of another, giving a new hybrid. In total, four hybrids are presented: backmarking with backjumping (BMJ), backmarking with conflict‐directed backjumping (BM‐CBJ), forward checking with backjumping (FC‐BJ), and forward checking with conflict‐directed backjumping (FC‐CBJ). The performances of the nine algorithms (BT, BJ, CBJ, BM, BMJ, BM‐CBJ, FC, FC‐BJ, FC‐CBJ) are compared empirically, using 450 instances of the ZEBRA problem, and it is shown that FC‐CBJ is by far the best of the algorithms examined.

An Empirical Study of Phase Transitions in Binary Constraint Satisfaction Problems

Abstract An empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables, domain size, and connectivity there is a critical level of constraint tightness at which a phase transition occurs. At the phase transition, problems change from being soluble to insoluble, and the difficulty of problems increases dramatically. A theory developed by Williams and Hogg [44], and independently developed by Smith [37], predicts where the hardest problems should occur. It is shown that the theory is in close agreement with the empirical results, except when constraint graphs are sparse.

Solving Vehicle Routing Problems Using Constraint Programming and Metaheuristics

Constraint Programming typically uses the technique of depth-first branch and bound as the method of solving optimization problems. Although this method can give the optimal solution, for large problems, the time needed to find the optimal can be prohibitive. This paper introduces a method for using local search techniques within a Constraint Programming framework, and applies this technique to vehicle routing problems. We introduce a Constraint Programming model for vehicle routing, and a system for integrating Constraint Programming and local search techniques. We then describe how the method can be accelerated by handling core constraints using fast local checks, while other more complex constraints are left to the constraint propagation system. We have coupled our local search method with a meta-heuristic to avoid the search being trapped in local minima. Several meta-heuristics are investigated ranging from a simple Tabu Search method to Guided Local Search. An empirical study over benchmark problems shows the relative merits of these techniques. Investigations indicate that the specific long-term memory technique used by Guided Local Search can be used as a diversification method for Tabu Search, resulting in significant benefit. Several new best solutions on the Solomon problems are found in relatively few iterations of our algorithm.

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.

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.

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.

Guided Local Search for the Vehicle Routing Problem with Time Windows

We describe a heuristic solution method for the Vehicle Routing Problem with Time Windows. The method uses four improvement operators in a steepest descent search strategy. The Guided Local Search meta-heuristic is used to avoid local minima. The resulting algorithm is tested on Solomon’s capacitated vehicle routing problems with time windows. We compare our results to the best heuristic approaches reported for the VRPTW. The new method performs significantly better than previous methods on classes where vehicle routes tend to be longer. The method is slightly worse on classes with shorter routes. We report 12 new best solutions for Solomon’s problems.

Exact Algorithms for Maximum Clique: A Computational Study

We take an existing implementation of an algorithm for the maximum clique problem and modify it so that we can distribute it over an ad-hoc cluster of machines. Our goal was to achieve a signicant speedup in performance with minimal development eort, i.e. a maximum costup. We present a simple modication to a state-of-the-art exact algorithm for maximum clique that allows us to distribute it across many machines. An empirical study over large hard benchmarks shows that speedups of an order of magnitude are routine for 25 or more machines.We investigate a number of recently reported exact algorithms for the maximum clique problem. The program code is presented and analyzed to show how small changes in implementation can have a drastic effect on performance. The computational study demonstrates how problem features and hardware platforms influence algorithm behaviour. The effect of vertex ordering is investigated. One of the algorithms (MCS) is broken into its constituent parts and we discover that one of these parts frequently degrades performance. It is shown that the standard procedure used for rescaling published results (i.e., adjusting run times based on the calibration of a standard program over a set of benchmarks) is unsafe and can lead to incorrect conclusions being drawn from empirical data.

A Comparison of Traditional and Constraint-based HeuristicMethods on Vehicle Routing Problems with Side Constraints

The vehicle routing problem (VRP) is a variantof the familiar travelling salesperson problem (TSP). In theVRP we are to perform a number of visits, using a number of vehiclesof limited capacity, while typically minimizing the distancetravelled. VRPs can be complicated by imposing time windows ordeadlines on visits, sequencing constraints between visits, andso on. In this paper, we use a constraint-based toolkit for solvingvehicle routing problems to study the effect of different heuristictechniques. We investigate the performance of a number of constructionand improvement techniques, and show that as the size of thesolution space is decreased through addition of side constraints,certain conventional techniques fail while constraint directedtechniques continue to perform acceptably. This suggests thatconstraint programming techniques are particularly suited toVRPs with side constraints.

A Constraint Programming Approach to the Stable Marriage Problem

The Stable Marriage problem (SM) is an extensively-studied combinatorial problem with many practical applications. In this paper we present two encodings of an instance I of SM as an instance J of a Constraint Satisfaction Problem. We prove that, in a precise sense, establishing arc consistency in J is equivalent to the action of the established Extended Gale/Shapley algorithm for SM on I. As a consequence of this, the man-optimal and woman-optimal stable matchings can be derived immediately. Furthermore we show that, in both encodings, all solutions of I may be enumerated in a failure-free manner. Our results indicate the applicability of Constraint Programming to the domain of stable matching problems in general, many of which are NP-hard.