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.

Contents 1 Introduction 2 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Problem Formulations 4 2.1 Integer Programming Formulations . . . . . . . . . . . . . . . . . . . 5 2.2 Continuous Formulations . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Computational Complexity 12 4 Bounds and Estimates 15 5 Exact Algorithms 19 5.1 Enumerative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Exact Algorithms for the Unweighted Case . . . . . . . . . . . . . . 21 5.3 Exact Algorithms for the Weighted Case . . . . . . . . . . . . . . . . 25 6 Heuristics 27 6.1 Sequential Greedy Heuristics . . . . . . . . . . . . . . . . . . . . . . 28 6.2 Local Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3 Advanced Search Heuristics . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.2 Neural networks . . . . . . . . . . . . . . . . . . . . . . . .

We present a method for solving the independent set formulation of the graph coloring problem (where there is one variable for each independent set in the graph). We use a column generation method for implicit optimization of the linear program at each node of the branch-and-bound tree. This approach, while requiring the solution of a difficult subproblem as well as needing sophisticated branching rules, solves small to moderate size problems quickly. We have also implemented an exact graph coloring algorithm based on DSATUR for comparison. Implementation details and computational experience are presented. 1 INTRODUCTION The graph coloring problem is one of the most useful models in graph theory. This problem has been used to solve problems in school timetabling [10], computer register allocation [7, 8], electronic bandwidth allocation [11], and many other areas. These applications suggest that effective algorithms for solving the graph coloring problem would be of great importance. D...

A new Reactive Local Search (RLS ) algorithm is proposed for the solution of the Maximum-Clique problem. RLS is based on local search complemented by a feedback (history-sensitive) scheme to determine the amount of diversification. The reaction acts on the single parameter that decides the temporary prohibition of selected moves in the neighborhood, in a manner inspired by Tabu Search. The performance obtained in computational tests appears to be significantly better with respect to all algorithms tested at the the second DIMACS implementation challenge. The worst-case complexity per iteration of the algorithm is O(max{n, m}) where n and m are the number of nodes and edges of the graph. In practice, when a vertex is moved, the number of operations tends to be proportional to its number of missing edges and therefore the iterations are particularly fast in dense graphs.

Given a graph, in the maximum clique problem one wants to find

vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1

this paper, it is shown how to take advantage of the properties of these models to approximately solve the maximum clique problem, a well-known intractable optimization problem which has practical applications in various fields. The approach is based on a result by Motzkin and Straus which naturally leads to formulate the problem in a manner that is readily mapped onto a relaxation labeling network. Extensive simulations have demonstrated the validity of the proposed model, both in terms of quality of solutions and speed. Maximum clique problem, relaxation labeling processes, neural networks, optimization. 1 INTRODUCTION

We report on experiments with a new hybrid graph coloring algorithm, which combines a parallel version of Morgenstern's S-Impasse algorithm [20], with exhaustive search. We contribute new test data arising in five different application domains, including register allocation and class scheduling. We test our algorithms both on this test data and on several types of randomly generated graphs. We compare our parallel implementation, which is done on the CM-5, with two simple heuristics, the Saturation algorithm of Br'elaz [4] and the Recursive Largest First (RLF) algorithm of Leighton [18]. We also compare our results with previous work reported by Morgenstern [20] and Johnson et al. [13]. Our main results are as follows. ffl On the randomly generated graphs, the performance of Hybrid is consistently better than the sequential algorithms, both in terms of speed and number of colorings produced. However, on large random graphs, our algorithms do not come close to the best colorings found ...

In Pattern Recognition, the Graph Matching problem involves the matching of a sample data graph with the subgraph of a larger model graph where vertices and edges correspond to pattern parts and their relations. In this paper, we present Rulegraphs, a new method that combines the Graph Matching approach with Rule-Based approaches from Machine Learning. This new method reduces the cardinality of the (NP-Complete) Graph Matching problem by replacing model part, and their relational, attribute states by rules which depict attribute bounds and evidence for di erent classes. We show how rulegraphs, when combined with techniques for checking feature label-compatibilities, not only reduce the search space but also improve the uniqueness of the matching process.

We explore neural network and related heuristic methods for the fast approximate solution of the Maximum Clique problem. One of these algorithms, Mean Field Annealing, is implemented on the Connection Machine CM-5 and a fast annealing schedule is experimentally evaluated on random graphs, as well as on several benchmark graphs. The other algorithms, which perform certain randomized local search operations, are evaluated on the same benchmark graphs, and on Sanchis graphs. One of our algorithms adjusts its internal parameters as its computation evolves. On Sanchis graphs, it finds significantly larger cliques than the other algorithms do. Another algorithm, GSD(;), works best overall, but is slower than the others. All our algorithms obtain significantly larger cliques than other simpler heuristics but run slightly slower; they obtain significantly smaller cliques on average than exact algorithms or more sophisticated heuristics but run considerably faster. All our algorithms are simple...