发表机构
ID Lab, University of Ghent-imec（根特大学-imec ID实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对分数背包问题中贪心规则对输入扰动敏感的问题，提出基于分组的两阶段分配模型，通过分组与比率排序降低损失，并保证连续性与高效计算。
AI 中文摘要
为解决分数背包问题，丹齐格贪心规则根据价值-成本比率的顺序对物品进行排序。这种排序引入了优先级问题。如果预算在具有非常相似比率的两个物品之间耗尽，输入的一个任意小扰动可能改变分配。为缓解该问题，我们引入一个两阶段规则。我们将具有在半径δ内共享属性的物品分组。然后按比率降序评估这些组，并在不进一步排序的情况下分配其组的预算份额。考虑一个具有总容量U_G、单位成本在[w^-,w^+]内、以及代表值v̂的组。该组相对于精确最优解的损失由v̂ U_G (w^+-w^-)/(w^++w^-)+ε_v U_G界定，其中ε_v限制了组内价值变化。此外，对于任何组大小，该调和因子保持紧致。只要分组保持顺序兼容，总体损失就被限制在唯一的预算约束组上；因此，包含至多K个物品的组每个物品损失为O(K/n)。如果组比率区间重叠至多ω，则附加项ωC会使该界限变差。在相邻组之间的分离余量内，分组分配相对于成本数据保持利普希茨连续，模量为K/w_min。计算该分配需要O(n+m log m+|Γ| log|Γ|)时间，给定m个组和一个边界组Γ。或者，如果线性时间选择方法确定边界组的分配，时间复杂度降至O(n+m log m)。
英文摘要
To solve the fractional knapsack problem, Dantzig's greedy rule orders items according to their value-to-cost ratio. This ordering introduces priority issues. An arbitrarily small perturbation to the input can change the allocation if the budget is exhausted between two items with very similar ratios. To mitigate that problem, we introduce a two-stage rule. We group items sharing attributes within a radius $δ$. We then evaluate these groups in descending order of ratio and divide their group's budget share without further ranking. Consider a group featuring an aggregate capacity $U_G$, unit costs contained in $[w^-,w^+]$, and a representative value $\widehat{v}$. The group's loss relative to the exact optimum is bounded by $\widehat{v}\, U_G\frac{w^+-w^-}{w^++w^-}+\varepsilon_v U_G$, where $\varepsilon_v$ limits the group's internal value variation. Moreover, this harmonic factor remains tight for any group size. The overall loss becomes restricted to the single budget-binding group whenever the grouping remains order-compatible; thus, groups containing at most $K$ items suffer a per-item loss of $\mathcal{O}(\frac{K}{n})$. Should group ratio intervals exhibit an overlap of at most $ω$, an additive term $ωC$ degrades this bound. Within the separation margin between adjacent groups, the grouped allocation remains Lipschitz continuous with respect to cost data, exhibiting a modulus of $\frac{K}{w_{\min}}$. Computing this allocation takes $\mathcal{O}(n+m\log m+|Γ|\log|Γ|)$ time given $m$ groups and a boundary group $Γ$. Alternatively, the time complexity drops to $\mathcal{O}(n+m\log m)$ if a linear-time selection method identifies the boundary group's allocation.
