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arXiv · 2607.13955

Pack, Remove, Reserve -- Online Knapsack with Second Thoughts

Hans-Joachim Böckenhauer·Dennis Komm·Emanuel Skodinis·Moritz Stocker·Philip Whittington
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Abstract

We study the online proportional knapsack problem with two paid forms of recourse. Items arrive one by one and must be handled immediately, without knowledge of the future: an algorithm may pack an item $x$, reject it, or reserve it for possible later use at proportional cost $\alpha x$; additionally, it may at any time remove previously packed items, at proportional cost $\beta y$ for each removed item $y$. Reservation and removal have each been analyzed in isolation, but their combination raises a natural question: is the better of the two mechanisms always optimal on its own, or is there a region in the parameter space spanned by $\alpha$ and $\beta$ in which they genuinely enter into a symbiosis? So far, this question has only been answered for the special case of free removal ($\beta = 0$), leaving the vast majority of the parameter space unexplored. We close this gap, determining matching upper and lower bounds on the competitive ratio for every pair of cost parameters $(\alpha, \beta)$ and revealing three qualitatively different regimes. In some regions, reservation alone already achieves the optimal ratio; in others, removal alone does. However, most interestingly, in the heart of the parameter space lies a symbiosis region in which combining both mechanisms is strictly better than either one on its own. The optimal algorithm in the symbiosis region is a natural blend of the two known single-mechanism strategies: postponing commitment by reserving until a threshold is reached, then packing greedily and revising via removal.

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Hans-Joachim Böckenhauer, Dennis Komm, Emanuel Skodinis, Moritz Stocker, Philip Whittington. 2026-07-15. Pack, Remove, Reserve -- Online Knapsack with Second Thoughts. https://arxiv.org/abs/2607.13955

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