We use a local search method we term Large Neighbourhood Search (LNS) for solving vehicle routing problems. LNS meshes well with constraint programming technology and is analogous to the shuffling technique of job-shop scheduling. The technique explores a large neighbourhood of the current solution by selecting a number of customer visits to remove from the routing plan, and re-inserting these visits using a constraint-based tree search. We analyse the performance of LNS on a number of vehicle routing benchmark problems. Unlike related methods, we use Limited Discrepancy Search during the tree search to re-insert visits. We also maintain diversity during search by dynamically altering the number of visits to be removed, and by using a randomised choice method for selecting visits to remove. We analyse the performance of our method for various parameter settings controlling the discrepancy limit, the dynamicity of the size of the removal set, and the randomness of the choice. We demonst...

This paper describes our experience with a simple modeling and programming approach for increasing the amount of constraint propagation in the constraint solving process. The idea, although similar to redundant constraints, is based on the concept of redundant modeling. We introduce the notions of CSP model and model redundancy, and show how mutually redundant models can be combined and connected using channeling constraints. The combined model contains the mutually redundant models as sub-models. Channeling constraints allow the sub-models to cooperate during constraint solving by propagating constraints freely amongst the sub-models. This extra level of pruning and propagation activities becomes the source of execution speedup. We perform two case studies to evaluate the effectiveness and efficiency of our method. The first case study is based on the simple and well-known n-queens problem, while the second case study applies our method in the design and construction of a real-life ...

. In the last twenty years, many algorithms and heuristics were developed to find solutions in constraint networks. Their number increased to such an extent that it quickly became necessary to compare their performances in order to propose a small number of "good" methods. These comparisons often led us to consider FC or FC-CBJ associated with a "minimum domain" variable ordering heuristic as the best techniques to solve a wide variety of constraint networks. In this paper, we first try to convince once and for all the CSP community that MAC is not only more efficient than FC to solve large practical problems, but it is also really more efficient than FC on hard and large random problems. Afterwards, we introduce an original and efficient way to combine variable ordering heuristics. Finally, we conjecture that when a good variable ordering heuristic is used, CBJ becomes an expensive gadget which almost always slows down the search, even if it saves a few constraint checks. 1 Introducti...

The performance of backtracking algorithms for solving finite-domain constraint satisfaction problems can be improved substantially by look-back and look-ahead methods. Look-back techniques extract information by analyzing failing search paths that are terminated by dead-ends. Look-ahead techniques use constraint propagation algorithms to avoid such dead-ends altogether. This survey describes a number of look-back variants including backjumping and constraint recording which recognize and avoid some unnecessary explorations of the search space. The last portion of the paper gives an overview of look-ahead methods such as forward checking and dynamic variable ordering, and discusses their combination with backjumping.

Over the past twenty veyears many backtracking algorithms have been developed for constraint satisfaction problems. This survey describes the basic backtrack search within the search space framework and then presents a number of improvements developed in the past two decades, including look-back methods such asbackjumping, constraint recording, backmarking, and look-ahead methods such as forward checking and dynamic variable ordering. 1

Abstraction and reformulation are fundamental, powerful ideas in artificial intelligence, but they have not had a great deal of application in the area of constraint satisfaction. The obvious way to implement abstraction in a constraint satisfaction context is to simplify the problem by removing constraints, and then use the solutions to the simplified problem to guide the search for a solution to the original problem. In a sense, local search, hill climbing methods use this approach, while giving up completeness guarantees. The problem for a complete method is that a simplified problem may be trivial (recent work on locating hard problems suggests that hard problems may cluster on narrow ridges in problem space). Thus there may be too many solutions to the simplified problem to be useful. The Cartesian product representation and interchangeability techniques provide ways of working with compact representations of large sets of solutions.

A constraint satisfaction problem (CSP) consists of a set of variables; for each variable, a finite set of possible values (its domain); and a set of constraints restricting the values that the variables can simultaneously take. A solution to a CSP is an assignment of a value from its domain to every variable, in such a way that every constraint is satisfied. Many problems arising in O.R., in particular scheduling, timetabling and other combinatorial problems, can be represented as CSPs. Constraint programming tools now exist which allow CSPs to be expressed easily, and provide standard strategies for finding solutions. This tutorial is intended to give a basic grounding in constraint satisfaction problems and some of the algorithms used to solve them, including the techniques commonly used in constraint programming tools. In particular, it covers arc and path consistency; simple backtracking and forward checking, as examples of search algorithms; and the use of heuristics to guide the...

When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. Consider, for example, permutation problems in which we have as many values as variables, and each variable takes an unique value. In such problems, we can choose between a primal and a dual viewpoint. In the dual viewpoint, each dual variable represents one of the primal values, whilst each dual value represents one of the primal variables. Alternatively, by means of channelling constraints to link the primal and dual variables, we can have a combined model with both sets of variables. In this paper, we perform an extensive theoretical and empirical study of such primal, dual and combined models for two classes of problems: permutation problems and injection problems. Our results show that it often be advantageous to use multiple viewpoints, and to have constraints which channel between them to maintain consistency. They also illustrate a general...

In this paper we describe a paradigm for contentfocused matchmaking, based on a recently proposed model for constraint acquisition-and-satisfaction. Matchmaking agents are conceived as constraintbased solvers that interact with other, possibly human, agents (Clients or Customers). The Matchmaker provides potential solutions ("suggestions") based on partial knowledge, while gaining further information about the problem itself from the other agent through the latter's evaluation of these suggestions. The dialog between Matchmaker and Customer results in iterative improvement of solution quality, as demonstrated in simple simulations. We also show empirically that this paradigm supports "suggestion strategies " for finding acceptable solutions more efficiently or for increasing the amount of information obtained from the Customer. This work also indicates some ways in which the tradeoff between these two metrics for evaluating performance can be handled. Introduction Intelligent matchm...

. We present a detailed experimental comparison of interleaved depth-first search and depth-bounded discrepancy search, two tree search procedures recently developed with the same goal: to reduce the cost of heuristic mistakes at the top of the tree. Our comparison uses an abstract heuristic model, and three different concrete problem classes: binary constraint satisfaction, quasigroup completion and number partitioning problems. Results indicate that both search strategies often reduce search. In addition, they show that their efficiency depends on a trade-off between the number of discrepancies (branch points against the heuristic) considered at the top of the tree, and the overhead of expanding branches from these discrepancies. If the number of discrepancies is large, the overhead can outweigh the benefits. 1 INTRODUCTION By definition, heuristics sometimes make mistakes. When searching a tree with depth-first search (Dfs), mistakes made at the top of the tree can be very costly ...