In resource allocation, we often require that the output allocation of an algorithm is stable against input perturbation because frequent reallocation is costly and untrustworthy. Varma and Yoshida (SODA'21) formalized this requirement for algorithms as the notion of average sensitivity. Here, the average sensitivity of an algorithm on an input instance is, roughly speaking, the average size of the symmetric difference of the output for the instance and that for the instance with one item deleted, where the average is taken over the deleted item. In this work, we consider the average sensitivity of the knapsack problem, a representative example of a resource allocation problem. We first show a $(1-\epsilon)$-approximation algorithm for the knapsack problem with average sensitivity $O(\epsilon^{-1}\log \epsilon^{-1})$. Then, we complement this result by showing that any $(1-\epsilon)$-approximation algorithm has average sensitivity $\Omega(\epsilon^{-1})$. As an application of our algorithm, we consider the incremental knapsack problem in the random-order setting, where the goal is to maintain a good solution while items arrive one by one in a random order. Specifically, we show that for any $\epsilon > 0$, there exists a $(1-\epsilon)$-approximation algorithm with amortized recourse $O(\epsilon^{-1}\log \epsilon^{-1})$ and amortized update time $O(\log n+f_\epsilon)$, where $n$ is the total number of items and $f_\epsilon>0$ is a value depending on $\epsilon$.

翻译：在资源分配中，我们常要求算法的输出分配能稳定应对输入扰动，因为频繁的重新分配不仅成本高昂且缺乏可靠性。Varma与Yoshida（SODA'21）将这一对算法的要求形式化为平均敏感度的概念。此处，算法在输入实例上的平均敏感度粗略而言是指：当删除一个物品时，原实例输出与删除后实例输出的对称差大小的平均值，该平均值在所有可能的被删物品上计算。本研究聚焦资源分配问题的典型代表——背包问题的平均敏感度。我们首先提出一个平均敏感度为$O(\epsilon^{-1}\log \epsilon^{-1})$的背包问题$(1-\epsilon)$近似算法。进而通过证明任何$(1-\epsilon)$近似算法均具有$\Omega(\epsilon^{-1})$的平均敏感度，对此结果形成理论补充。作为算法的应用，我们探讨了随机序设定下的增量背包问题，其目标是在物品按随机顺序逐一到达时持续维持优质解。具体而言，我们证明对于任意$\epsilon > 0$，存在具有$O(\epsilon^{-1}\log \epsilon^{-1})$分摊调整次数与$O(\log n+f_\epsilon)$分摊更新时间的$(1-\epsilon)$近似算法，其中$n$为物品总数，$f_\epsilon>0$为依赖于$\epsilon$的数值。