Celso C. Ribeiro

Reactive GRASP: An Application to a Matrix Decomposition Problem in TDMA Traffic Assignment

A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. In this paper, we describe a GRASP for a matrix decomposition problem arising in the context of traffic assignment in communication satellites. We review basic concepts of GRASP: construction and local search algorithms. The local search phase is based on the use of a new type of neighborhood defined by constructive and destructive moves. The implementation of a GRASP for the matrix decomposition problem is described in detail. Extensive computational experiments on literature and randomly generated problems are reported. Moreover, we propose a new procedureReactive GRASP, in which the basic parameter that defines the restrictiveness of the candidate list during the construction phase is self-adjusted according to the quality of the solutions previously found. The approach is robust and does not require calibration efforts. On most of the literature problems considered, the newReactive GRASP heuristic matches the optimal solution found by an exact column-generation with branch-and-bound algorithm.

GRASP with Path-Relinking: Recent Advances and Applications

Path-relinking is a major enhancement to the basic greedy randomized adaptive search procedure (GRASP), leading to significant improvements in solution time and quality. Path-relinking adds a memory mechanism to GRASP by providing an intensification strategy that explores trajectories connecting GRASP solutions and the best elite solutions previously produced during the search. This paper reviews recent advances and applications of GRASP with path-relinking. A brief review of GRASP is given. This is followed by a description of path-relinking and how it is incorporated into GRASP. Several recent applications of GRASP with path-relinking are reviewed. The paper concludes with a discussion of extensions to this strategy, concerning in particular parallel implementations and applications of path-relinking with other metaheuristics.

Experimental and Efficient Algorithms

The multiprocessor scheduling problem consists in scheduling a set of tasks with known processing times into a set of identical processors so as to minimize their makespan, i.e., the maximum processing time over all processors. We propose a new heuristic for solving the multiprocessor scheduling problem, based on a hybrid heuristic to the bin packing problem. Computational results illustrating the effectiveness of this approach are reported and compared with those obtained by other heuristics.

Invited Review: Scheduling in sports: An annotated bibliography

Sports have worldwide appeal. Professional sport leagues involve significant investments in players. Events such as the Olympics Games, the Football World Cup and the major golf and tennis tournaments generate huge worldwide television audiences and many sports are multi-million dollar industries. A key aspect of sporting events is the ability to generate schedules that optimize logistic issues and that are seen as fair to all those who have an interest. This is not just restricted to generating the fixtures, but also to other areas such as assigning officials to the games in the competitions. This paper provides an annotated bibliography for sports scheduling articles. This area can be traced back over 40 years. It is noticeable that the number of papers has risen in recent years, demonstrating that scientific interest is increasing in this area.

A Column Generation Approach to the Multiple-Depot Vehicle Scheduling Problem

We give a new formulation to the multiple-depot vehicle scheduling problem as a set partitioning problem with side constraints, whose continuous relaxation is amenable to be solved by column generation. We show that the continuous relaxation of the set partitioning formulation provides a much tighter lower bound than the additive bound procedure previously applied to this problem. We also establish that the additive bound technique cannot provide tighter bounds than those obtained by Lagrangian decomposition, in the framework in which it has been used so far. Computational results that illustrate the robustness of the combined set partitioning-column generation approach are reported for problems four to five times larger than the largest problems that have been exactly solved in the literature. Finally, we show that the gap associated with the additive bound based on the assignment and shortest path relaxations can be arbitrarily bad in the general case, and as bad as 50% in the symmetric case.

A Hybrid GRASP with Perturbations for the Steiner Problem in Graphs

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 multi-start approach, the greedy randomized construction phase of a GRASP is replaced by the use of several construction heuristics with a weight perturbation strategy that combines intensification and diversification elements, as in a strategic oscillation approach. The improvement phase circularly explores two different local search strategies. The first uses anode-based neighborhood for local search, while the second uses a key-path-based neighborhood. An adaptive path-relinking technique is applied to a set of elite solutions as apost-optimization strategy. Computational results on a broad set of benchmark problems illustrate the effectiveness and the robustness of our heuristic, which is very competitive when compared to other approximate algorithms.

Randomized heuristics for the Max-Cut problem

Given an undirected graph with edge weights, the MAX-CUT problem consists in finding a partition of the nodes into two subsets, such that the sum of the weights of the edges having endpoints in different subsets is maximized. It is a well-known NP-hard problem with applications in several fields, including VLSI design and statistical physics. In this article, a greedy randomized adaptive search procedure (GRASP), a variable neighborhood search (VNS), and a path-relinking (PR) intensification heuristic for MAX-CUT are proposed and tested. New hybrid heuristics that combine GRASP, VNS, and PR are also proposed and tested. Computational results indicate that these randomized heuristics find near-optimal solutions. On a set of standard test problems, new best known solutions were produced for many of the instances.

Probability Distribution of Solution Time in GRASP: An Experimental Investigation

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 approximately achieve linear speed-up by implementing GRASP in parallel.

Greedy Randomized Adaptive Search Procedures: Advances, Hybridizations, and Applications

GRASP is a multi-start metaheuristic for combinatorial optimization problems, in which each iteration consists basically of two phases: construction and local search. The construction phase builds a feasible solution, whose neighborhood is investigated until a local minimum is found during the local search phase. The best overall solution is kept as the result. In this chapter, we first describe the basic components of GRASP. Successful implementation techniques are discussed and illustrated by numerical results obtained for different applications. Enhanced or alternative solution construction mechanisms and techniques to speed up the search are also described: alternative randomized greedy construction schemes, Reactive GRASP, cost perturbations, bias functions, memory and learning, local search on partially constructed solutions, hashing, and filtering. We also discuss implementation strategies of memory-based intensification and post-optimization techniques using path-relinking. Hybridizations with other metaheuristics, parallelization strategies, and applications are also reviewed.

TTT plots: a perl program to create time-to-target plots

This paper describes a perl language program to create time-to-target solution value plots for measured CPU times that are assumed to fit a shifted exponential distribution. This is often the case in local search based heuristics for combinatorial optimization, such as simulated annealing, genetic algorithms, iterated local search, tabu search, WalkSAT, and GRASP. Such plots are very useful in the comparison of different algorithms or strategies for solving a given problem and have been widely used as a tool for algorithm design and comparison. We first discuss how TTT plots are generated. This is followed by a description of the perl program tttplots.pl.