Problems involving the efficient arrangement of simple objects, as captured by bin packing and makespan scheduling, are fundamental tasks in combinatorial optimization. These are well understood in the traditional online and offline cases, but have been less well-studied when the volume of the input is truly massive, and cannot even be read into memory. This is captured by the streaming model of computation, where the aim is to approximate the cost of the solution in one pass over the data, using small space. As a result, streaming algorithms produce concise input summaries that approximately preserve the optimum value. We design the first efficient streaming algorithms for these fundamental problems in combinatorial optimization. For Bin Packing, we provide a streaming asymptotic (1 + ε)-approximation with \(\widetilde {O}\)\(\left (\frac {1}{\varepsilon }\right )\), where \(\widetilde {{{O}}}\) hides logarithmic factors. Moreover, such a space bound is essentially optimal. Our algorithm implies a streaming (d + ε)-approximation for Vector Bin Packing in d dimensions, running in space \(\widetilde {{{O}}}\left (\frac {d}{\varepsilon }\right )\). For the related Vector Scheduling problem, we show how to construct an input summary in space \(\widetilde {{{O}}}(d^{2}\cdot m / \varepsilon ^{2})\) that preserves the optimum value up to a factor of \(2 - \frac {1}{m} +\varepsilon \), where m is the number of identical machines.

由装箱打包和makepan调度捕获的涉及简单对象的有效排列的问题是组合优化中的基本任务。在传统的联机和脱机情况下，这些都是很好理解的，但是当输入量确实很大，甚至无法读入内存时，对它们的研究就很少。这是通过流计算模型来捕获的，该模型的目的是使用较小的空间在数据上传递一次以估算解决方案的成本。结果，流算法产生简洁的输入摘要，大致保留了最佳值。我们针对组合优化中的这些基本问题设计了第一个有效的流算法。对于Bin Packing，我们提供了流渐近（1 + ε）-通过\（\ widetilde {O} \）\（\ left（\ frac {1} {\ varepsilon} \ right} \）逼近，其中\（\ widetilde {{{O}}} \）隐藏了对数因子。此外，这样的空间界限本质上是最佳的。我们的算法意味着流（d + ε）为-近似向量装箱在d的尺寸，在空间中运行\（\ widetilde {{{ö}}} \左（\压裂{d} {\ varepsilon} \右）\ ）。对于相关的向量调度问题，我们展示了如何在空间\（\ widetilde {{{O}}}（d ^ {2} \ cdot m / \ varepsilon ^ {2}）\）中构造输入摘要，以保留最佳值，最高可达\（2-\ frac {1} {m} + \ varepsilon \），其中m是相同机器的数量。