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 $3+2\sqrt{2}\approx 5.83$.