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

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, Bezakova and Dani (SIGecom Exch 5(3):11-18, 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 each item j is either heavy (w(j) = 1) or light (w(j) = epsilon), for some parameter epsilon is an element of (0, 1). We show that the (1, epsilon)-restricted case is also 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 root 2 approximate to 5.83.