变尺寸广义箱覆盖的近似算法
Approximation Algorithms for Variable-Sized and Generalized Bin Covering
摘要：
我们考虑广义箱覆盖（GBC）问题：我们给出$m$箱类型，其中每个箱的类型$i$有利润$p_i$和需求$d_i$。此外，还有$n$项，其中项$j$的大小为$s_j$。如果分配给它的项目集的总大小至少为需求$d_i$，则覆盖$i$类型的箱。在这种情况下，$p_i$的利润被赚取，目标是使总利润最大化。据我们所知，以前仅治疗过$p_i = d_i = 1$（箱覆盖）和$p_i = d_i$（可变尺寸箱覆盖（VSBC））病例。我们研究了两种仓位供应模型：在单位供应模型中，每种类型只有一个仓位，即\，例如，我们有独立的垃圾箱。相比之下，在无限供给模型中，每种类型都有任意多个箱子。显然，单位供给模型是无限供给模型的推广。据我们所知，单位供应模型还没有研究。我们的单位供给模型的结果不仅渐近，但所有的情况下。这与以前关于\prob{Bin Covering}的大多数工作形成了对比。我们证明了存在一个具有运行时间$\bigO{nm\sqrt{m+n}}$的单位供给GBC的组合5-近似算法。此外，对于VSBC，我们表明，自然和快速的下一个适合减少（$\NFD$）算法是一个9/4近似的单位供应模型。该算法的界是紧的，接近最佳可能。我们证明了在\emph{infinite}供给模型中存在VSBC的AFPTAS。
英文摘要：
We consider the Generalized Bin Covering (GBC) problem: We are given $m$ bin types, where each bin of type $i$ has profit $p_i$ and demand $d_i$. Furthermore, there are $n$ items, where item $j$ has size $s_j$. A bin of type $i$ is covered if the set of items assigned to it has total size at least the demand $d_i$. In that case, the profit of $p_i$ is earned and the objective is to maximize the total profit. To the best of our knowledge, only the cases $p_i = d_i = 1$ (Bin Covering) and $p_i = d_i$ (Variable-Sized Bin Covering (VSBC)) have been treated before. We study two models of bin supply: In the unit supply model, we have exactly one bin of each type, i.\,e., we have individual bins. By contrast, in the infinite supply model, we have arbitrarily many bins of each type. Clearly, the unit supply model is a generalization of the infinite supply model. To the best of our knowledge the unit supply model has not been studied yet. Our results for the unit supply model hold not only asymptotically, but for all instances. This contrasts most of the previous work on \prob{Bin Covering}. We prove that there is a combinatorial 5-approximation algorithm for GBC with unit supply, which has running time $\bigO{nm\sqrt{m+n}}$. Furthermore, for VSBC we show that the natural and fast Next Fit Decreasing ($\NFD$) algorithm is a 9/4-approximation in the unit supply model. The bound is tight for the algorithm and close to being best-possible. We show that there is an AFPTAS for VSBC in the \emph{infinite} supply model.
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