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DOI: 10.4230/LIPIcs.ISAAC.2016.23
URN: urn:nbn:de:0030-drops-67939
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6793/
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Chan, T-H. Hubert ; Tang, Zhihao Gavin ; Wu, Xiaowei

On (1, epsilon)-Restricted Max-Min Fair Allocation Problem

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Abstract

We study the max-min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight w_j. For this setting, Asadpour et al. [TALG, 2012] showed that a certain configuration-LP can be used to estimate the optimal value within a factor of 4 + delta, for any delta > 0, which was recently extended by Annamalai et al. [SODA 2015] to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani [SIGecom Exch., 2005] showed that it is NP-hard to approximate the problem within any ratio smaller than 2. In this paper we consider the (1, epsilon)-restricted max-min fair allocation problem, in which for some parameter epsilon in (0, 1), each item j is either heavy (w_j = 1) or light (w_j = epsilon). We show that the (1, epsilon)-restricted case is also NP-hard to approximate within any ratio smaller than 2. Hence, this simple special case is still algorithmically interesting. Using the configuration-LP, we are able to estimate the optimal value of the problem within a factor of 3 + delta, for any delta > 0. Extending this idea, we also obtain a quasi-polynomial time (3 + 4 epsilon)-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as epsilon tends to 0, the approximation ratio of our polynomial-time algorithm approaches 3 + 2 sqrt{2} approx 5.83.

BibTeX - Entry

@InProceedings{chan_et_al:LIPIcs:2016:6793,
  author =	{T-H. Hubert Chan and Zhihao Gavin Tang and Xiaowei Wu},
  title =	{{On (1, epsilon)-Restricted Max-Min Fair Allocation Problem}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{23:1--23:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Seok-Hee Hong},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6793},
  URN =		{urn:nbn:de:0030-drops-67939},
  doi =		{10.4230/LIPIcs.ISAAC.2016.23},
  annote =	{Keywords: Max-Min Fair Allocation, Hypergraph Matching}
}

Keywords: Max-Min Fair Allocation, Hypergraph Matching
Seminar: 27th International Symposium on Algorithms and Computation (ISAAC 2016)
Issue Date: 2016
Date of publication: 02.12.2016


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