Random Graphs and Graph Optimization Problems

One major difficulty in analyzing algorithms for graph optimization problems is that the probabilistic behavior of the optimum solutions to most of the important problems is generally unknown. We present a general method for relating some well-known results regarding the probability of existence of certain subgraphs in random graphs to the probabilistic behavior of solutions to graph optimization problems, where the problem graphs have edge weights independently chosen from an arbitrary distribution. Application of the technique to well-studied problems such as the traveling salesman problem shows that stronger statements can be made about the optimum solutions than have previously been proved, and that the analysis is straightforward.