ANALYSIS OF HEURISTIC FOR NUMBER PARTITIONINGIan

We illustrate the use of phase transition behaviour in the study of heuristics. Using an \an-nealed" theory, we deene a parameterthat measures the \constrainedness"of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show that constrainedness can be used to analyse and compare algorithms and heuristics for number partitioning in a precise and quantitative manner. For example, we demonstrate that on uniform random problems both the Karmarkar-Karp and greedy heuristics minimize the con-strainedness, but that the decisions made by the Karmarkar-Karp heristic are superior at reducing constrainedness. This supports the better performance observed experimentally for the Karmarkar-Karp heuristic. Our results refute a conjecture of Y. Fu that phase transition behaviour does not occur in number partitioning. In addition, they demonstrate that phase transition behaviour is useful for more than just simple benchmarking. It can, for instance, be used to analyse heuristics, and to compare the quality of heuristic solutions.