An Asymptotical Study of Combinatorial Optimization Problems by Means of Statistical Mechanics

The analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study. Simulated annealing, a metaheuristic for combinatorial optimization problems, is based on this analogy. In this paper we use the statistical mechanics formalism based on the above mentioned analogy to analyze the asymptotic behavior of a special class of combinatorial optimization problems characterized by a combinatorial conditions which is well known in the literature. Our result is analogous to results of other authors derived by purely probabilistic means: Under natural prob-abilistic conditions on the coeecients of the problem, the ratio between the optimal value and size of a feasible solution approaches almost surely the expected value of the coeecients, as the size of the problem tends to innnity. Our proof shows clearly why the above mentioned combinatorial condition which characterizes the class of investigated problems is essential.