Mauricio G. C. Resende

Greedy Randomized Adaptive Search Procedures

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.

Designing and reporting on computational experiments with heuristic methods

This article discusses the design of computational experiments to test heuristic methods and provides reporting guidelines for such experimentation. The goal is to promote thoughtful, well-planned, and extensive testing of heuristics, full disclosure of experimental conditions, and integrity in and reproducibility of the reported results.

A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set

An efficient randomized heuristic for a maximum independent set is presented. The procedure is tested on randomly generated graphs having from 400 to 3,500 vertices and edge probabilities from 0.2 to 0.9. The heuristic can be implemented trivially in parallel and is tested on an MIMD computer with 1, 2, 4 and 8 processors. Computational results indicate that the heuristic frequently finds the optimal or expected optimal solution in a fraction of the time required by implementations of simulated annealing, tabu search, and an exact partial enumeration method.

A Hybrid Heuristic for the p -Median Problem

Given n customers and a set F of m potential facilities, the p-median problem consists in finding a subset of F with p facilities such that the cost of serving all customers is minimized. This is a well-known NP-complete problem with important applications in location science and classification (clustering). We present a multistart hybrid heuristic that combines elements of several traditional metaheuristics to find near-optimal solutions to this problem. Empirical results on instances from the literature attest the robustness of the algorithm, which performs at least as well as other methods, and often better in terms of both running time and solution quality. In all cases the solutions obtained by our method were within 0.1% of the best known upper bounds.

FEEDBACK SET PROBLEMS

Not long ago, there appeared to be a consensus in the literature that feedback set problems, which originated from the area of combinational circuit design, were the least understood among all the classical combinatorial optimization problems due to the lack of positive results in efficient exact and approximating algorithms. This picture has been totally changed in recent years. Dramatic progress has occurred in developing approximation algorithms with provable performance; new bounds have been established one after the other and it is probably fair to say that feedback set problems are becoming among the most exciting frontend problems in combinatorial optimization.

A continuous approach to inductive inference

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean functionℱ:{0, 1}n → {0, 1} using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input—output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used for defining the search region is dynamically modified. Computational results on 8-, 16- and 32-input, 1-output functions are presented. Our implementation successfully identified the majority of hidden functions in the experiment.

A Parallel Grasp for the Steiner Tree Problem in Graphs Using a Hybrid Local Search Strategy

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 neighborhood search strategies is then proposed. Computational experiments with a parallel implementation of the hybrid procedure are reported, showing that the algorithm found optimal solutions for 45 out of 60 benchmark instances and was never off by more than 4% of the optimal solution value. The average speedup results observed for the test problems show that increasing the number of processors reduces elapsed times with increasing speedups. Moreover, the main contribution of the parallel algorithm concerns the fact that larger speedups of the same order of the number of processors are obtained exactly for the most difficult problems.

Parallel Search for Combinatorial Optimization: Genetic Algorithms, Simulated Annealing, Tabu Search and GRASP

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.

Computing Lower Bounds for the Quadratic Assignment Problem with an Interior Point Algorithm for Linear Programming

An example of the quadratic assignment problem QAP is the facility location problem, in which n facilities are assigned, at minimum cost, to n sites. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the product of the flow and the distance between sites to which the facilities are assigned. Proving optimality of QAPs has been limited to instances having fewer than 20 facilities, largely because known lower bounds are weak. We compute lower bounds for a wide range of QAPs using a linear programming-based lower bound studied by Z. Drezner 1995. On the majority of QAPs tested, a new best known lower bound is computed. On 87% of the instances, we produced the best known lower bound. On several instances, including some having more the 20 facilities, the lower bound is tight. The linear programs, which can be large even for small QAPs, are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the interior point directions. Attempts to solve these instances using the CPLEX primal simplex algorithm as well as the CPLEX barrier primal-dual interior point method were successful only for the smallest instances.

Computational experience with an interior point algorithm on the satisfiability problem

We apply the zero-one integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large instances of SAT. Hundreds of instances of SAT (having from 100 to 1000 variables and 100 to 32,000 clauses) are randomly generated and solved. For comparison, we attempt to solve the problems via linear programming relaxation with MINOS.