Random Knapsacks with Many Constraints

We provide new results on asymptotic values for the random knapsack problem. For a very general model in which the parameters are determined by a rather arbitrary joint distribution, we compute the rate of growth as the number of objects increases, the number of constraints being fixed. For a particular model, we find strong bounds on the asymptotic value as the numbers of objects and constraints increase together. This paper is a continuation of the work in [3,4] on estimating the values of random knapsack problems with many decision variables. It consists of two independent parts. In Section 1, we show how to estimate the growth rate of the value of a random knapsack when the parameters are determined by a very general class of joint distributions. In Section 2, we concentrate on a particular random knapsack model, and give rather sharp new bounds on its asymptotic value. In more detail: In Section 1, we first settle a question left open in [3] related to a single-constraint random knapsack problem, then apply this new result to a multiconstraint problem. Consider the problem V, = max i XjSj, j= 1 subject to i ~~j I K, djc (03 l} j=l where the random variable pairs ( Wj, Xi) are independent, identically distributed draws from any one of a very wide class of joint distributions F,,. (In particular, we do not assume that W and X are independent.) For t > 0, let F(t) = E( W 1 ix t rW;) and G(t) = E(X 1 ix 2 tw;). Correspondence to: Professor K. Schilling, Department of Mathematics, University of Michigan-Flint, Flint, MI 48502.2186, USA. 0166-218X/94/$07.00