On ($1$, $\epsilon$)-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. 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. to give a polynomial-time $13$-approximation algorithm for the problem. For hardness results, Bezakova and Dani 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 $\epsilon \in (0,1)$. 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 $3+2\sqrt{2}\approx 5.83$.