The benefit of adaptivity in the stochastic knapsack problem with dependence on the state of nature

Abstract

We consider a stochastic variant of the NP-hard 0/1 knapsack problem in which item values are deterministic and item sizes are random variables with known arbitrary distributions. These distributions depend on a common random variable θ denoting the state of the nature. We assume that if we fix a value of θ, then the size variables are independent. So θ induces a limited dependency among the sizes of different items. 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 that maximizes the expected value of items successfully placed in the knapsack, where the final overflowing item contributes no value. We consider both non-adaptive policies (that designate a priori a fixed permutation of items to insert) and adaptive policies (that can make dynamic decisions based on the instantiated sizes of 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 study this measure as a function of the cardinality of the support of θ. Assuming that the support of θ has at most k values, we show a lower bound of Ω(k) and an upper bound of O(k) on the adaptivity gap in our model. We also introduce Ω(lnk) lower bound and O(lnk) upper bound for the case, where the following two additional assumptions hold. The first assumption is stochastic monotonicity of the sizes in terms of θ and the second is that the prior distribution of θ is uniform. We show that both assumptions are vital, i.e., one assumption without the other does not bring us to a sub-linear adaptivity gap. We further show that in the last O(lnk) upper bound on the price of adaptivity we cannot replace the assumption of stochastic monotonicity with the weaker assumption that the item sizes are positively correlated.