scholarly journals Streaming Algorithms for Bin Packing and Vector Scheduling

2020 ◽  
pp. 72-88 ◽  
Author(s):  
Graham Cormode ◽  
Pavel Veselý
Keyword(s):  
Bin Packing ◽  
Streaming Algorithms

scholarly journals Streaming Algorithms for Bin Packing and Vector Scheduling

2020 ◽  
Author(s):  
Graham Cormode ◽  
Pavel Veselý
Keyword(s):  
Combinatorial Optimization ◽  
Bin Packing ◽  
Streaming Algorithms ◽  
Small Space ◽  
Model Of Computation ◽  
Identical Machines ◽  
The Cost

AbstractProblems 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 $\widetilde {O}$ O ~ $\left (\frac {1}{\varepsilon }\right )$ 1 ε , where $\widetilde {{{O}}}$ 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 $\widetilde {{{O}}}\left (\frac {d}{\varepsilon }\right )$ O ~ d ε . For the related Vector Scheduling problem, we show how to construct an input summary in space $\widetilde {{{O}}}(d^{2}\cdot m / \varepsilon ^{2})$ O ~ ( d 2 ⋅ m / ε 2 ) that preserves the optimum value up to a factor of $2 - \frac {1}{m} +\varepsilon $ 2 − 1 m + ε , where m is the number of identical machines.

Tight Bounds for Clairvoyant Dynamic Bin Packing

2019 ◽  
Vol 6 (3) ◽  
pp. 1-21 ◽  
Author(s):  
Yossi Azar ◽  
Danny Vainstein
Keyword(s):  
Bin Packing

Power of d Choices for Large-Scale Bin Packing

2015 ◽  
Vol 43 (1) ◽  
pp. 321-334 ◽  
Author(s):  
Qiaomin Xie ◽  
Xiaobo Dong ◽  
Yi Lu ◽  
Rayadurgam Srikant
Keyword(s):  
Large Scale ◽  
Bin Packing

scholarly journals Dynamic Windows Scheduling with Reallocation

2021 ◽  
Vol 26 ◽  
pp. 1-19
Author(s):  
Martín Farach-Colton ◽  
Katia Leal ◽  
Miguel A. Mosteiro ◽  
Christopher Thraves Caro
Keyword(s):  
Supply Chain ◽  
Bin Packing ◽  
Time Slot ◽  
Peak Load ◽  
Communication Channels ◽  
Current Load ◽  
Constant Amount ◽  
Unit Fractions ◽  
Consecutive Time ◽  
Multiple Variants

We consider the Windows Scheduling (WS) problem, which is a restricted version of Unit-Fractions Bin Packing, and it is also called Inventory Replenishment in the context of Supply Chain. In brief, WS problem is to schedule the use of communication channels to clients. Each client c i is characterized by an active cycle and a window w i . During the period of time that any given client c i is active, there must be at least one transmission from c i scheduled in any w i consecutive time slots, but at most one transmission can be carried out in each channel per time slot. The goal is to minimize the number of channels used. We extend previous online models, where decisions are permanent, assuming that clients may be reallocated at some cost. We assume that such cost is a constant amount paid per reallocation. That is, we aim to minimize also the number of reallocations. We present three online reallocation algorithms for Windows Scheduling. We evaluate experimentally multiple variants of these protocols showing that, in practice, all three achieve constant amortized reallocations with close to optimal channel usage. Our simulations also expose interesting tradeoffs between reallocations and channel usage. We introduce a new objective function for WS with reallocations that can be also applied to models where reallocations are not possible. We analyze this metric for one of the algorithms that, to the best of our knowledge, is the first online WS protocol with theoretical guarantees that applies to scenarios where clients may leave and the analysis is against current load rather than peak load. Using previous results, we also observe bounds on channel usage for one of the algorithms.

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