Lower bounds on the adaptivity gaps in variants of the stochastic knapsack problem

We consider stochastic variants of the NP-hard 0/1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. The goal is to compute a policy for insertion of the items, that maximizes the expected value of the set of items placed in the knapsack. These variants that we study differ only in the formula for computing the value of the final solution obtained by the policy. We consider both nonadaptive policies (that designate a priori a fixed subset or permutation of items to insert) and adaptive policies (that can make dynamic decisions based on the instantiated sizes of the items placed in the knapsack thus far). Our work characterizes the benefit of adaptivity. For this purpose we use a measure called the adaptivity gap: the supremum over instances of the ratio between the expected value obtained by an optimal adaptive policy and the expected value obtained by an optimal non-adaptive policy. We show that while for the variants considered in the literature this quantity is bounded by a constant there are other variants where it is unbounded.