On the growth of random knapsacks

where the pairs (I+$, Xi) 2 0 are assumed to be independent draws from a common . . . . . joint distribution Fwx. If we think of the pairs (w,Xi) as the weights and values, respectively, of a collection of n objects, then this problem can be thought of as finding the collection of objects of maximum value which will fit in a “knapsack” with weight capacity 1. Our main result, Theorem 2.4, computes the asymptotic value of the random variables I$, with increasing n, for a very large class of joint distributions Fwx. Frieze and Clarke [2] computed the asymptotic value of this random knapsack problem for a particular F,, (where Wand X are mutually independent and both uniformly distributed on the interval (0,l)) as part of their analysis of approximation algorithms for the deterministic knapsack problem. In a related paper, Meante, Rinnooy Kan, Stougie, and Vercellis [3] analyze a random knapsack problem in which the knapsack capacity grows in proportion to the number n of items. Under this assumption, v,/n converges. Among their results, Meante et al. compute this limit. In contrast, when the knapsack capacity is fixed the growth rate of V,