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 wj . For this setting, Asadpour et al. [2] showed that a certain configuration-LP can be used to estimate the optimal value within a factor of 4+δ, for any δ > 0, which was recently extended by Annamalai et al. [1] to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani [5] showed that it is NP-hard to approximate the problem within any ratio smaller than 2. In this paper we consider the (1, )-restricted max-min fair allocation problem, in which for some parameter ∈ (0, 1), each item j is either heavy (wj = 1) or light (wj = ). We show that the (1, )-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 + δ, for any δ > 0. Extending this idea, we also obtain a quasi-polynomial time (3 + 4 )-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as tends to 0, the approximation ratio of our polynomial-time algorithm approaches 3 + 2 √ 2 ≈ 5.83. 1998 ACM Subject Classification G.1.2 Approximation, G.1.6 Optimization, G.2.1 Combinatorics