Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning

Abstract

This is the second in a series of three papers that empirically examine the competitiveness of simulated annealing in certain well-studied domains of combinatorial optimization. Simulated annealing is a randomized technique proposed by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi for improving local optimization algorithms. Here we report on experiments at adapting simulated annealing to graph coloring and number partitioning, two problems for which local optimization had not previously been thought suitable. For graph coloring, we report on three simulated annealing schemes, all of which can dominate traditional techniques for certain types of graphs, at least when large amounts of computing time are available. For number partitioning, simulated annealing is not competitive with the differencing algorithm of N. Karmarkar and R. M. Karp, except on relatively small instances. Moreover, if running time is taken into account, natural annealing schemes cannot even outperform multiple random runs of the local optimization algorithms on which they are based, in sharp contrast to the observed performance of annealing on other problems.

The Vertex Coloring Problem and its generalizations

“…This suggests that effective algorithms would be of great importance. Despite its relevance, few exact algorithms for VCP have been proposed, and are able to solve consistently only small instances, with up to 100 vertices for random graphs [72,101,103,41]. On the other hand, several heuristic and metaheuristic algorithms have been proposed which are able to deal with graphs of hundreds or thousands of vertices.…”

“…They are mainly based on simulated annealing (Johnson, Aragon, McGeoch and Schevon [72] compared different neighborhoods and presented extensive computational results on random graphs; Morgenstern [92] proposed a very effective neighborhood search) or Tabu Search (Hertz and De Werra [62]; Dorne and Hao [43]; Caramia and Dell'Olmo [25] proposed a local search with priorities rules, inspired from Tabu Search techniques). Funabiki and Higashino [50] proposed one of the most effective algorithms for the problem, which combines a Tabu Search technique with different heuristic procedures, color fixing and solution recombination in the attempt to expand a feasible partial coloring to a complete coloring.…”

A Refined Evaluation Function for the MinLA Problem

“…However, this method can not be used if the search space includes infeasible solutions. In such cases, penalty terms are often added to evaluate the degree of infeasibility [3,8]. It is also effective to dynamically change the evaluation function during the search, like in the noising method [1] and the search space smoothing method [6].…”

“…It is also effective to dynamically change the evaluation function during the search, like in the noising method [1] and the search space smoothing method [6]. Another technique consists in developing new more informative evaluation functions which may not be directly related to the objective function such in [8,9].…”

Parallel Simulated Annealing Algorithm for Graph Coloring Problem

“…Most algorithms designed for GCP are iterative heuristics [11], such as genetic algorithms [6], simulated annealing [7,8], tabu or local search techniques [9], minimizing selected cost functions. At the time of this writing, the only parallel metaheuristic for GCP is parallel genetic algorithm [10,[12][13][14][15].…”