Logarithmic regret in the dynamic and stochastic knapsack problem
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We study a dynamic and stochastic knapsack problem in which a decision maker is sequentially presented with n items with unitary rewards and independent weights that are drawn from a known continuous distribution F. The decision maker seeks to maximize the expected number of items that she includes in the knapsack while satisfying a capacity constraint, and while making terminal decisions as soon as each item weight is revealed. Under mild regularity conditions on the weight distribution F, we prove that the regret---the expected difference between the performance of the best sequential algorithm and that of a prophet who sees all of the weights before making any decision---is, at most, logarithmic in n. Our proof is constructive. We devise a re-optimized heuristic that achieves this regret bound.
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