Routing with time windows (VRPTW) has been an area of research that have attracted many researchers within the last 10 { 15 years. In this period a number of papers and technical reports have been published on the exact solution of the VRPTW. The VRPTW is a generalization of the well-known capacitated routing problem (VRP or CVRP). In the VRP a eet of vehicles must visit (service) a number of customers. All vehicles start and end at the depot. For each pair of customers or customer and depot there is a cost. The cost denotes how much is costs a vehicle to drive from one customer to another. Every customer must be visited exactly ones. Additionally each customer demands a certain quantity of goods delivered (know as the customer demand). For the vehicles we have an upper limit on the amount of goods that can be carried (known as the capacity). In the most basic case all vehicles are of the same type and hence have the same capacity. The problem is now for a given scenario to plan routes for the vehicles in accordance with the mentioned constraints such that the cost accumulated on the routes, the xed costs (how much does it cost to maintain a vehicle) or a combination hereof is minimized. In the more general VRPTW each customer has a time window, and between all pairs of customers or a customer and the depot we have a travel time. The vehicles now have to comply with the additional constraint that servicing of the customers can only be started within the time windows of the customers. It is legal to arrive before a time window \opens" but the vehicle must wait and service will not start until the time window of the customer actually opens. For solving the problem exactly 4 general types of solution methods have evolved in the literature: dynamic programming, Dantzig-Wolfe (column generation), Lagrange decomposition and solving the classical model formulation directly. Presently the algorithms that uses Dantzig-Wolfe given the best results (Desrochers, Desrosiers and Solomon, and Kohl), but the Ph.D. thesis of Kontoravdis shows promising results for using the classical model formulation directly. In this Ph.D. project we have used the Dantzig-Wolfe method. In the Dantzig-Wolfe method the problem is split into two problems: a \master problem" and a \subproblem". The master problem is a relaxed set partitioning v vi problem that guarantees that each customer is visited exactly ones, while the subproblem is a shortest path problem with additional constraints (capacity and time window). Using the master problem the reduced costs are computed for each arc, and these costs are then used in the subproblem in order to generate routes from the depot and back to the depot again. The best (improving) routes are then returned to the master problem and entered into the relaxed set partitioning problem. As the set partitioning problem is relaxed by removing the integer constraints the solution is seldomly integral therefore the Dantzig-Wolfe method is embedded in a separation-based solution-technique. In this Ph.D. project we have been trying to exploit structural properties in order to speed up execution times, and we have been using parallel computers to be able to solve problems faster or solve larger problems. The thesis starts with a review of previous work within the eld of VRPTW both with respect to heuristic solution methods and exact (optimal) methods. Through a series of experimental tests we seek to dene and examine a number of structural characteristics. The rst series of tests examine the use of dividing time windows as the branching principle in the separation-based solution-technique. Instead of using the methods previously described in the literature for dividing a problem into smaller problems we use a methods developed for a variant of the VRPTW. The results are unfortunately not positive. Instead of dividing a problem into two smaller problems and try to solve these we can try to get an integer solution without having to branch. A cut is an inequality that separates the (non-integral) optimal solution from all the integer solutions. By nding and inserting cuts we can try to avoid branching. For the VRPTW Kohl has developed the 2-path cuts. In the separationalgorithm for detecting 2-path cuts a number of test are made. By structuring the order in which we try to generate cuts we achieved very positive results. In the Dantzig-Wolfe process a large number of columns may be generated, but a signicant fraction of the columns introduced will not be interesting with respect to the master problem. It is a priori not possible to determine which columns are attractive and which are not, but if a column does not become part of the basis of the relaxed set partitioning problem we consider it to be of no benet for the solution process. These columns are subsequently removed from the master problem. Experiments demonstrate a signicant cut of the running time. Positive results were also achieved by stopping the route-generation process prematurely in the case of time-consuming shortest path computations. Often this leads to stopping the shortest path subroutine in cases where the information (from the dual variables) leads to \bad" routes. The premature exit from the shortest path subroutine restricts the generation of \bad" routes signi cantly. This produces very good results and has made it possible to solve problem instances not solved to optimality before. The parallel algorithm is based upon the sequential Dantzig-Wolfe based algorithm developed earlier in the project. In an initial (sequential) phase unsolved problems are generated and when there are unsolved problems enough vii to start work on every processor the parallel solution phase is initiated. In the parallel phase each processor runs the sequential algorithm. To get a good workload a strategy based on balancing the load between neighbouring processors is implemented. The resulting algorithm is eÆcient and capable of attaining good speedup values. The loadbalancing strategy shows an even distribution of work among the processors. Due to the large demand for using the IBM SP2 parallel computer at UNIC it has unfortunately not be possible to run as many tests as we would have liked. We have although managed to solve one problem not solved before using our parallel algorithm.

In this paper we present a parallel implementation of a Greedy Randomized Adaptive Search Procedure (grasp) for finding approximate solutions to the quadratic assignment problem. In particular, we discuss efficient techniques for large-scale sparse quadratic assignment problems on an MIMD parallel computer. We report computational experience on a collection of quadratic assignment problems. The code was run on a Kendall Square Research KSR-1 parallel computer, using 1, 4, 14, 24, 34, 44, 54, and 64 processors, and achieves an average speedup that is almost linear in the number of processors. 1 Introduction Nonlinear assignment problems, such as quadratic, cubic, and N-adic assignment problems were formulated by Lawler [11]. One of the most extensively studied nonlinear assignment problems is the quadratic assignment problem (QAP). The QAP was first introduced by Koopmans and Beckmann in 1957 as a mathematical model for locating a set of indivisible economic activities [9]. Consider th...

Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.

There have been proposed efficient ways of enumerating all the association rules that are interesting with respect to support, confidence, or other measures. In contrast, we examine the optimization problem of computing the optimal association rule that maximizes the significance of the correlation between the assumption and the conclusion of the rule. We propose a parallel branch-and-bound graph search algorithm tailored to this problem. The key features of the design are (1) novel branchand -bound heuristics, and (2) a rule of rewriting conjunctions that avoids maintaining the list of visited nodes. Experiments on two different types of large-scale shared-memory multi-processors confirm that the speed-up of the computation time scales almost linearly with the number of processors, and the size of search space could be dramatically reduced by the branch-and-bound heuristics. 1 Introduction Many organizations are seeking strategies for processing or interpreting massive amounts of da...

. In this paper, we review parallel metaheuristics for approximating the global optimal solution of combinatorial optimization problems. Recent developments on parallel implementation of genetic algorithms, simulated annealing, tabu search, variable neighborhood search, and greedy randomized adaptive search procedures (GRASP) are discussed. 1. Introduction Search techniques are fundamental problem-solving methods in computer science and operations research. Search algorithms have been used to solve many classes of problems, including path-finding problems, two-player games, and constraint satisfaction problems. Classical examples of path-finding problems include many combinatorial optimization problems (e.g. integer programming) and puzzles (e.g. Rubic's cube, Eight Puzzle). Chess, backgammon, and Othello belong to the class of two player games, while a classic example of a constraint satisfaction problem is the eight-queens problem. In this paper, we focus on NP-hard combinator...

Abstract. The weighted maximum satis ability (MAX-SAT) problem is central in mathematical logic, computing theory, and many industrial applications. In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for solving MAX-SAT problems. Experimental results indicate that almost linear speedup is achieved.

.We analyze and compare the scalabilityoftwo generic schemes for heuristic depth-#rst search on highly parallel MIMD systems. The #rst one employs a task attraction mechanism where the work packets are generated on demand by splitting the donor's stack. Analytical and empirical analyses show that this stack-splitting scheme works e#ciently on parallel systems with a small communication diameter and a moderate number of processing elements. The second scheme, search-frontier splitting, also employs a task attraction mechanism, but uses pre-computed work packets taken from a search-frontier level of the tree. At the beginning, a search-frontier is generated and stored in the local memories. Then, the processors expand the subtrees of their frontier nodes, communicating only when they run out of work or a solution has been found. Empirical results obtained on a 32 # 32 = 1024 node MIMD system indicate that the search-frontier splitting scheme incurs fewer overheadsand scale...

In discrete optimization problems, we search for an optimal solution among all vectors in a discrete solution space that satisfy a set of constraints, and the search efficiency is of considerable importance since exhaustive search is often impracticable. The method called branch-andbound (noted B&B) is a heuristic tree search algorithm used in this context. Its principle lies in successive decompositions of the original problem in smaller disjoint subproblems until an optimal solution is found, and the search avoids visiting some subproblems which are known not to contain an optimal solution. Given that disjoint subproblems can be decomposed simultaneously and independently, parallel processing has been widely considered as an additional source of improvement in search efficiency, using the set of processors to concurrently decompose several subproblems at each iteration. Parallel B&B is traditionally considered as an irregular parallel algorithm due to the fact that the structure o...

Abstract. In this paper, we review parallel search techniques for approximating the global optimal solution of combinatorial optimization problems. Recent developments on parallel implementation of genetic algorithms, simulated annealing, tabu search, and greedy randomized adaptive search procedures (GRASP) are discussed.

Discrete optimization problems (DOPs) arise in various applications such as planning, scheduling, computer aided design, robotics, game playing and constraint directed reasoning. Often, a DOP is formulated in terms of finding a (minimum cost) solution path in a graph from an initial node to a goal node and solved by graph/tree search methods. Availability of parallel computers has created substantial interest in exploring parallel formulations of these graph and tree search methods. This article provides a survey of various parallel search algorithms such as Backtracking, IDA*, A*, Branch-and-Bound techniques and Dynamic Programming. It addresses issues related to load balancing, communication costs, scalability and the phenomenon of speedup anomalies in parallel search.