Average Case Analysis of the Set Packing Problem
The paper deals with the well known set packing problem and its special case, when the number of subsets is maximized. It is assumed that some of the problem coeﬃcients are realizations of mutually independent random variables. Average case (i.e. asymptotical probabilistic) properties of selected problem characteristics are investigated for the variety of possible instances of the problem. The important results of the paper are: Behavior of the optimal solution values of the set packing problem is presented for the special asymptotic case, where mutual asymptotical relation between m (number of elements of the packed set) and n (number of sets provided) is playing an essential role. Probability of reaching feasible solution is reasonably high (i.e. > 2/e,2/e ≈ 0.736); moreover, it may be set arbitrarily close to 1 (e.g. 0.999), although the deterioration in the quality of approximation of the behavior of the optimal solution values may be substantial.Some relations between the general case of the set packing problem and its maximization for the special case are investigated. (original abstract)
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