# A Competitive Analysis of Online Knapsack Problems with Unit Density

We study an online knapsack problem where the items arrive sequentially and must be either immediately packed into the knapsack or irrevocably discarded. Each item has a different size and the objective is to maximize the total size of items packed. While the competitive ratio of deterministic algorithms for this problem is known to be 0, the competitive ratio of randomized algorithms has, surprisingly, not been considered until now. We derive a random-threshold algorithm which is 0.432-competitive, and show that our threshold distribution is optimal. We also consider the generalization to multiple knapsacks, where an arriving item has a different size in each knapsack and must be placed in at most one. This is equivalent to the Adwords problem where item truncation is not allowed. We derive a randomized algorithm for this problem which is 0.214-competitive.

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