Georgia Institute of Technology College of Sciences

We examine a variant of the knapsack problem in which item sizes are random according to an arbitrary but known distribution. In each iteration, an item size is realized once the decision maker chooses and attempts to insert an item. With the aim of maximizing the expected profit, the process ends when either all items are successfully inserted or a failed insertion occurs. We investigate the strength of a particular dynamic programming based LP bound by examining its gap with the optimal adaptive policy. Our new relaxation is based on a quadratic value function approximation which introduces the notion of diminishing returns by encoding interactions between remaining items. We compare the bound to previous bounds in literature, including the best known pseudopolynomial bound, and contrast their corresponding policies with two natural greedy policies. Our main conclusion is that the quadratic bound is theoretically more efficient than the pseudopolyomial bound yet empirically comparable to it in both value and running time.