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...

4, and Toby Walsh 5

. 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 randomlygenerated binary CSP's. 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 id...

. There has been considerable research interest into the solubility phase transition, and its effect on search cost for backtracking algorithms. In this paper we show that a similar easy-hard-easy pattern occurs for local search, with search cost peaking at the phase transition. This is despite problems beyond the phase transition having fewer solutions, which intuitively should make the problems harder to solve. We examine the relationship between search cost and number of solutions at different points across the phase transition, for three different local search procedures, across two problem classes (CSP and SAT). Our findings show that there is a significant correlation, which changes as we move through the phase transition. Keywords: computational complexity, constraint satisfaction, propositional satisfiability, search 1 Introduction Local search has been proposed as a good candidate for solving the "hard" but soluble problems that turn up at the phase transition in solubility f...

. We show that the same methodology used to study phase transition behaviour in NP-complete problems works with a polynomial problem class: establishing arc consistency. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NP-complete problems, predicts the location of a phase transition in establishing arc consistency. A complexity peak for the AC3 algorithm is associated with this transition. Finite size scaling models both the scaling of this transition and the computational cost. On problems at the phase transition, this model of computational cost agrees with the theoretical worst case. As with NP-complete problems, constrainedness -- and proxies for it which are cheaper to compute -- can be used as a heuristic for reducing the number of checks needed to establish arc consistency in AC3. 1 Introduction Following [4] there has been considerable research into phase transition behaviour in NP-complete problems. Problems from...

We show that phase transition behavior similar to that observed in NP-complete problems like random 3-Sat occurs further up the polynomial hierarchy in problems like random 2-Qsat. The differences between Qsat and Sat in phase transition behavior that Cadoli et al report are largely due to the presence of trivially unsatisfiable problems. Once they are removed, we see behavior more familiar from Sat and other NP-complete domains. There are, however, some differences. Problems with short clauses show a large gap between worst case behavior and median, and the easy-hard-easy pattern is restricted to higher percentiles of search cost. We compute

Many problem ensembles exhibit a phase transition that is associated with a large peak in the average cost of solving the problem instances. However, this peak is not necessarily due to a lack of solutions: indeed the average number of solutions is typically exponentially large. Here, we study this situation within the context of the satisfiability transition in Random 3SAT. We find that a significant subclass of instances emerges as we cross the phase transition. These instances are characterized by having about 85--95% of their variables occurring in unary prime implicates (UPIs), with their remaining variables being subject to few constraints. In such instances the models are not randomly distributed but all lie in a cluster that is exponentially large, but still admits a simple description. Studying the effect of UPIs on the local search algorithm Wsat shows that these "single-cluster" instances are harder to solve, and we relate their appearance at the phase transition to the peak...

The theoretical properties of qualitative spatial reasoning in the RCC-8 framework have been analyzed extensively. However, no empirical investigation has been made yet. Our experiments show that the adaption of the algorithms used for qualitative temporal reasoning can solve large RCC-8 instances, even if they are in the phase transition region -- provided that one uses the maximal tractable subsets of RCC-8 that have been identified by us. In particular, we demonstrate that the orthogonal combination of heuristic methods is successful in solving almost all apparently hard instances in the phase transition region up to a certain size in reasonable time.

Variable ordering heuristics can have a profound effect on the performance of backtracking search algorithms for constraint satisfaction problems. The smallest-remaining-domain heuristic is a commonlyused dynamic variable ordering heuristic, used in conjunction with algorithms such as forward checking which look ahead at the effects of each variable instantiation on those variables not yet instantiated. This heuristic has been explained as an implementation of the failfirst principle, stated by Haralick and Elliott [7], i.e. that the next variable selected should be the one which is most likely to result in an immediate failure. We calculate the probability that a variable will fail when using the forward checking algorithm to solve a class of binary CSPs. We derive a series of heuristics, starting with smallest-remaining-domain, based on increasingly accurate estimates of this probability, and predict that if the fail-first principle is sound, the more accurate the estimate the better...

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