Parallel Randomized Heuristics For The Set Covering Problem MARIA STELLA FIORENZO CATALANO Transportation and Traffic Engineering Section Delft University of Technology P.O. Box 5048 - 2600 GA Delft THE NETHERLANDS S.Catalano@CiTG.TUDelft.NL FEDERICO MALUCELLI DEI - Politecnico di Milano Piazza L. da Vinci 32 - 20133 Milano ITALY malucell@elet.polimi.it http://www.elet.polimi.it/people/malucell Abstract: - We propose a general scheme to derive heuristics for the Set Covering Problem. The scheme is iterative and embeds constructive heuristics within a randomized procedure. A first group of heuristics is obtained by randomizing the choices made at each step when the solution is constructed in a way similar to that of the so called "Ant System"; a second group of more efficient heuristics is obtained by introducing a random perturbation of the costs of the problem instance. Some computational results are presented. Different parallel implementations are discussed and some performance measures reported. Keywords: - heuristic algorithms, randomized methods, set covering, parallel algorithms 1. Introduction The Set Covering Problem plays an important role in combinatorial optimization. Many important applications can be formulated in terms of Set Covering, as for example crew scheduling, vehicle routing, facility location, assembly line balancing, information retrieval [10]. Given a set I = {1, , m} and a collection of proper subsets of I, F={I 1 , , I n }, a cover S ˝ F of I is such that each element of I belongs at least to one of the subsets in S. If we associate a cost c j with each subset I j ˛ F , and we define the cost of a cover as the sum of the cost of its components, the Set Covering Problem consists in finding minimum cost cover. Due to the difficulty of determining the optimal solution (the problem is NP-hard), and due to the great scale of real life problems, a many heuristic algorithms have been devised. Though it has been proved that the existence of a polynomial algorithm approximating the optimal solution within 1/4 log m , implies P=NP [20], in practice most of the literature heuristics very efficiently yield near optimal solutions [9]. In the present paper we propose a class of randomized heuristic algorithms. The scheme of the algorithms is iterative. At each iteration a feasible solution is constructed; this is done by exploiting any constructive heuristic of the literature. The construction of solutions considers not only the usual goodness criteria, but also the story of previously produced solutions, following the ideas introduced in the so called "Ant System" [12]. We will study the combination of this general scheme with some of the most efficient classical heuristics. The class of Randomized algorithms (with or without memory) proved to give good results when applied to the set covering in a large collection of test problems, as reported in [18]. In Section 2 we introduce the basic notation and we review some of the most efficient constructive heuristic algorithms to be used within the new heuristics. Section 3 illustrates the randomized heuristic scheme as well as some of the algorithms which can be obtained. Section 4 contains some computational results. The randomized heuristic scheme is easy to parallelize. In Section 5 we discuss some parallel implementations of the algorithms and we present the performance evaluation of selected heuristics.