License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
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DOI: 10.4230/LIPIcs.ISAAC.2020.11
URN: urn:nbn:de:0030-drops-133555
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/13355/
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Hyatt-Denesik, Dylan ; Rahgoshay, Mirmahdi ; Salavatipour, Mohammad R.

Approximations for Throughput Maximization

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Abstract

In this paper we study the classical problem of throughput maximization. In this problem we have a collection J of n jobs, each having a release time r_j, deadline d_j, and processing time p_j. They have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their [r_j,d_j] window. This problem has been studied extensively (even for the case of m = 1). Several special cases of the problem remain open. Bar-Noy et al. [STOC1999] presented an algorithm with ratio 1-1/(1+1/m)^m for m machines, which approaches 1-1/e as m increases. For m = 1, Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation with ratio 1-1/e-ε (for any ε > 0). Recently Im-Li-Moseley [IPCO2017] presented an algorithm with ratio 1-1/e+ε₀ for some absolute constant ε₀ > 0 for any fixed m. They also presented an algorithm with ratio 1-O(√(log m/m))-ε for general m which approaches 1 as m grows. The approximability of the problem for m = O(1) remains a major open question. Even for the case of m = 1 and c = O(1) distinct processing times the problem is open (Sgall [ESA2012]). In this paper we study the case of m = O(1) and show that if there are c distinct processing times, i.e. p_j’s come from a set of size c, then there is a randomized (1-ε)-approximation that runs in time O(n^{mc⁷ε^(-6)}log T), where T is the largest deadline. Therefore, for constant m and constant c this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning.

BibTeX - Entry

@InProceedings{hyattdenesik_et_al:LIPIcs:2020:13355,
  author =	{Dylan Hyatt-Denesik and Mirmahdi Rahgoshay and Mohammad R. Salavatipour},
  title =	{{Approximations for Throughput Maximization}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Yixin Cao and Siu-Wing Cheng and Minming Li},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13355},
  URN =		{urn:nbn:de:0030-drops-133555},
  doi =		{10.4230/LIPIcs.ISAAC.2020.11},
  annote =	{Keywords: Scheduling, Approximation Algorithms, Throughput Maximization}
}

Keywords: Scheduling, Approximation Algorithms, Throughput Maximization
Collection: 31st International Symposium on Algorithms and Computation (ISAAC 2020)
Issue Date: 2020
Date of publication: 04.12.2020

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