A probabilistic analysis of the set packing problem

Abstract

The maximum cardinality Set Packing Problem (SPP), formulated as max{e_nx: Ax≦e_m, xɛ{0,1}ⁿ}, where A is a mxn binary matrix and e_n, e_m are vectors of 1's of appropriate size, is a well-known NP-hard integer programming problem arising in a number of real-word applications.

In this paper, a probabilistic analysis of the SPP is developed. considering a stochastic model of the incidence matrix A in which the entries are independent Bernoulli distributed random variables, each with probability p_n of being equal to 1. A threshold function t_k (n,p_n) on the number m of constraints is derived for the property that a packing of cardinality k exists with probability tending to one as n tends to infinity.

In particular, it is shown that, if the probability p_n is constant, then the optimum value is almost surely equal to 1; thus combining the latter result with the corresponding one for the SCP obtained in [4], the duality gap is analyzed and shown to be asymptotically large as log n in ratio almost surely.

Finally, in § 4, the performance of the simple “blind” sequential algorithm is investigated, and a sufficient condition is assigned which the sequences p_n and m_n have to satisfy for the ratio of the optimal solution value to the approximate one to be asymptotically bounded by 2 almost surely.