Statistical mechanics of combinatorial search

The statistical mechanics of combinatorial search problems is described using the example of the well-known NP-complete graph coloring problem. A simple parameter describing the problem structure predicts the difficulty of solving the problem, on average. However, because of the large variance associated with this prediction, it is of limited direct use for individual instances. Additional parameters, describing the problem structure as well as the heuristic effectiveness, are introduced to address this issue. This also highlights the distinction between the statistical mechanics of combinatorial search problems, with their exponentially large search spaces, and physical systems, whose interactions are often governed by a simple Euclidean metric