AI 中文总结
研究具有两种付费形式的在线比例背包问题，确定每对成本参数\((\alpha, \beta)\)竞争比率的上下界，揭示三种情况，发现参数空间核心有共生区域，其最优算法是两种单机制策略融合。
AI 中文摘要
我们研究了具有两种付费形式的在线比例背包问题。物品逐一到达且必须立即处理，算法可以打包物品 \(x\)、拒绝它或按比例成本 \(\alpha x\) 预留以备后用；还可随时按比例成本 \(\beta y\) 移除已打包物品。此前对预留和移除单独分析过，但其结合引发问题：两种机制中更好的那个单独使用时是否总是最优，或者在由 \(\alpha\) 和 \(\beta\) 构成的参数空间中是否存在共生区域？目前仅在免费移除（\(\beta = 0\)）的特殊情况下有答案。我们填补了这一空白，确定了每对成本参数 \((\alpha, \beta)\) 的竞争比率的匹配上下界，并揭示了三种定性不同的情况。有些区域仅预留就可达到最优比率；有些区域仅移除可以。最有趣的是，在参数空间核心存在一个共生区域，结合两种机制比单独使用任何一种都严格更好。共生区域的最优算法是两种已知单机制策略的自然融合：通过预留推迟决策直到达到阈值，然后贪婪打包并通过移除修正。
英文摘要
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 $αx$; additionally, it may at any time remove previously packed items, at proportional cost $βy$ 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 $α$ and $β$ in which they genuinely enter into a symbiosis? So far, this question has only been answered for the special case of free removal ($β= 0$), 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 $(α, β)$ 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.
