Abstract. Today, a variety of heuristic approaches are available to the operations research practitioner. One methodology that has a strong intuitive appeal, a prominent empirical track record, and is trivial to efficiently implement on parallel processors is GRASP (Greedy Randomized Adaptive Search Procedures). GRASP is an iterative randomized sampling technique in which each iteration provides a solution to the problem at hand. The incumbent solution over all GRASP iterations is kept as the final result. There are two phases within each GRASP iteration: the first intelligently constructs an initial solution via an adaptive randomized greedy function; the second applies a local search procedure to the constructed solution in hope of finding an improvement. In this paper, we define the various components comprising a GRASP and demonstrate, step by step, how to develop such heuristics for combinatorial optimization problems. Intuitive justifications for the observed empirical behavior of the methodology are discussed. The paper concludes with a brief literature review of GRASP implementations and mentions two industrial applications.

. A GRASP (greedy randomized adaptive search procedure) is a multi-start metaheuristic for combinatorial optimization. We study the probability distributions of solution time to a sub-optimal target value in five GRASPs that have appeared in the literature and for which source code is available. The distributions are estimated by running 12,000 independent runs of the heuristic. Standard methodology for graphical analysis is used to compare the empirical and theoretical distributions and estimate the parameters of the distributions. We conclude that the solution time to a sub-optimal target value fits a two-parameter exponential distribution. Hence, it is possible to achieve linear speed-up by implementing GRASP in parallel. 1. Introduction A greedy randomized adaptive search procedure (GRASP) [8, 9, 11] is a multistart or iterative process, in which each GRASP iteration consists of two phases. In a construction phase, a feasible solution is produced and in a local search phase, a loc...

t We propose and describe a hybrid GRASP with weight perturbations and adaptive path-relinking heuristic (HGP+PR) for the Steiner problem in graphs. In this...

In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs. GRASP is a two phase metaheuristic. In the first phase, solutions are constructed using a greedy randomized procedure. Local search is applied in the second phase, leading to a local minimum with respect to a specified neighborhood. In the Steiner problem in graphs, feasible solutions can be characterized by their non-terminal nodes (Steiner nodes) or by their key-paths. According to this characterization, two GRASP procedures are described using different local search strategies. Both use an identical construction procedure. The first uses a node-based neighborhood for local search, while the second uses a path-based neighborhood. Computational results comparing the two procedures show that while the node-based variant produces better quality solutions, the path-based variant is about twice as fast. A hybrid GRASP procedure combining the two neighbo...

. GRASP (greedy randomized adaptive search procedure) is a metaheuristic for combinatorial optimization. GRASP usually is implemented as a multistart procedure, where each iteration is made up of a construction phase, where a randomized greedy solution is constructed, and a local search phase which starts at the constructed solution and applies iterative improvement until a locally optimal solution is found. This chapter gives an overview of GRASP. Besides describing the basic building blocks of a GRASP, the chapter covers enhancements to the basic procedure, including reactive GRASP, hybrid GRASP, and intensification strategies. 1. Introduction Consider a combinatorial optimization problem, where one is given a discrete set X of solutions and an objective function f(x) : x # X # to be minimized and seeks a solution x # # X such that f(x # ) # f(x), for all x # X . Problems of this type are sometimes easy to solve, i.e. they can be solved in polynomial time, but mor...

. Given an undirected graph with weights associated with its edges, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of nodes (terminals) of the original graph. We describe a reactive tabu search with path relinking algorithm for the Steiner problem in graphs, based on the extension of a previously developed tabu search algorithm using a neighborhood defined by insertions and eliminations of Steiner nodes. Computational experiments on benchmark problems are reported, comparing the reactive tabu search with other metaheuristic implementations. The reactive tabu search algorithm outperforms other algorithms, obtaining better or comparably good solutions. We also describe a robust parallel implementation based on an independent multiple path strategy and report improved computational results on a 32-processor cluster running under Linux. Key words. Combinatorial optimization, Steiner problem, graphs, metaheuristics, tabu search, reactive tabu...

. 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...

A GRASP (Greedy Randomized Adaptive Search Procedure) is a metaheuristic for producing good-quality solutions of combinatorial optimization problems. It is usually implemented with a construction procedure based on a greedy randomized algorithm followed by local search. In this Chapter, we survey parallel implementations of GRASP. We describe simple strategies to implement independent parallel GRASP heuristics and more complex cooperative schemes using a pool of elite solutions to intensify the search process. Some applications of independent and cooperative parallelizations are presented in detail.

A Greedy Randomized Adaptive Search Procedure (GRASP) is a metaheuristic for combinatorial optimization. It usually consists of a construction procedure based on a greedy randomized algorithm and a local search. Path-relinking is an intensification strategy that explores trajectories that connect high quality solutions. We analyze two parallel strategies for GRASP with path-relinking and propose a criterion to predict parallel speedup based on experiments with a sequential implementation of the algorithm. Independent and cooperative parallel strategies are described and implemented for the 3-index assignment problem and the job-shop scheduling problem. The computational results for independent parallel strategies are shown to qualitatively behave as predicted by the criterion.

. Given an undirected graph with weights associated with its nodes, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of nodes (terminals) of the original graph. In this paper, we describe an improved tabu search algorithm for the Steiner problem in graphs, based on a neighborhood defined by insertions and eliminations of Steiner nodes. Move estimations, elimination tests, and neighborhood reduction techniques are used to speedup the local search, leading to a very fast algorithm with very good results in terms of solution quality. Computational experiments on benchmark problems are reported, comparing the behavior of the improved tabu search with that of other heuristics from the literature. The improved tabu search algorithm clearly outperforms other heuristics in terms of computation times, obtaining better or comparably good solutions. Key words. Combinatorial optimization, Steiner problem, graphs, local search, metaheuristics, tabu sear...