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

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. (ACM Trans Algorithms 8(3):24, 2012) showed that a certain configuration-LP can be used to estimate the optimal value to within a factor of \(4+\delta \), for any \(\delta >0\), which was recently extended by Annamalai et al. (in: Indyk (ed) Proceedings of the twenty-sixth annual ACMSIAM symposium on discrete algorithms, SODA 2015, San Diego, CA, USA, January 4–6, 2015) to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani (SIGecom Exch 5(3):11–18, 2005) showed that it is \(\mathsf {NP}\)-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 each item j is either heavy \((w_j = 1)\) or light \((w_j = \epsilon )\), for some parameter \(\epsilon \in (0,1)\). We show that the \((1,\epsilon )\)-restricted case is also \(\mathsf {NP}\)-hard to approximate within any ratio smaller than 2. Using the configuration-LP, we are able to estimate the optimal value of the problem to 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 \(3+2\sqrt{2}\approx 5.83\).

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Correspondence to T.-H. Hubert Chan.

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A preliminary version of this paper appeared in the 27th International Symposium on Algorithms and Computation (ISAAC 2016). This research is supported in part by the Hong Kong RGC Grant 17202715.

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Chan, TH.H., Tang, Z.G. & Wu, X. On (1, \(\epsilon \))-Restricted Max–Min Fair Allocation Problem. Algorithmica 80, 2181–2200 (2018). https://doi.org/10.1007/s00453-018-0407-8

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Keywords

  • Max–min fair allocation
  • Hypergraph matching
  • Integrality gap