Abstract
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.
Highlights
The streaming model captures many scenarios when we must process very large volumes of data, which cannot fit into the working memory
While there have been many effective streaming algorithms designed for a range of problems in statistics, optimization, and graph algorithms, there has been little attention paid to the core problems of packing and scheduling
We present the first efficient algorithms for packing and scheduling that work in the streaming model
Summary
The streaming model captures many scenarios when we must process very large volumes of data, which cannot fit into the working memory. The algorithm makes one or more passes over the data with a limited memory, but does not have random access to the data It needs to extract a concise summary of the huge input, which can be used to approximately answer the problem under consideration. While there have been many effective streaming algorithms designed for a range of problems in statistics, optimization, and graph algorithms (see surveys by Muthukrishnan [39] and McGregor [38]), there has been little attention paid to the core problems of packing and scheduling. These are fundamental abstractions, which form the basis of many generalizations and extensions [13, 14]. We present the first efficient algorithms for packing and scheduling that work in the streaming model
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