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

Authors T-H. Hubert Chan, Zhihao Gavin Tang, Xiaowei Wu

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T-H. Hubert Chan
Zhihao Gavin Tang
Xiaowei Wu

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T-H. Hubert Chan, Zhihao Gavin Tang, and Xiaowei Wu. On (1, epsilon)-Restricted Max-Min Fair Allocation Problem. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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.
  • Max-Min Fair Allocation
  • Hypergraph Matching


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