🤖 AI Summary
This paper studies the online unit-density knapsack problem: heterogeneous-sized items arrive sequentially and must be irrevocably accepted or rejected upon arrival (no splitting), aiming to maximize total accepted size. The problem models real-time inventory allocation under unpredictable order streams in wholesale supply chains. We propose the first systematic framework for stochastic competitive ratio analysis and design an optimal randomized threshold policy: for a single knapsack, it achieves a tight 0.432-competitive ratio—matching the theoretical optimum; extended to multiple knapsacks, it attains a 0.214-competitive ratio—the best-known and tight bound to date. Our results establish the first nontrivial stochastic lower bound for the untruncated AdWords problem and significantly advance the theoretical frontier of online knapsack-type problems.
📝 Abstract
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.
