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.

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1 There may be more than one largest clique so we say we find "a largest clique" 2 At least two of the published algorithms name the candidate set P , maybe for "potential" vertices, for example [8] and more recently [7]. 3 We use the same name for this method as Tomita [26,27].

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We describe a heuristic solution method for the Vehicle RoutingProblem 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. INTRODUCTIONThe Vehicle Routing Problem (VRP) is a practical problem that has been studied widely in the OR literature [7]. The problem is usually expressed as follows: given a set of customers requiring a visit, and a fleet of vehicles based at a depot that can perform the visits, construct a set of routes for the vehicles which minimizes the costs of operation. The objective function is usually expressed as costs related to the number of vehicles used and to distance travelled.Vehicle routing problems generally come at least with capacity constraints on the vehicles (indicating that they can only carry a certain quantity of goods). However, other constraints are common. In this paper we examine the Vehicle Routing Problem with Time Windows (VRPTW). In this problem, as well as

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.

Abstract. The subgraph isomorphism problem involves finding a pattern graph inside a target graph. We present a new bit-and thread-parallel constraint-based search algorithm for the problem, and experiment on a wide range of standard benchmark instances to demonstrate its effectiveness. We introduce supplemental graphs, to create implied constraints. We use a new low-overhead, lazy variation of conflict directed backjumping which interacts safely with parallel search, and a counting-based all-different propagator which is better suited for large domains.