We consider the Max-Min Allocation problem, in which we are given a set of m agents and a set of n items, together with utilities u(A,i) of agent A for item i. Our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of its utilities for items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention recently, its approximability has not been well-understood thus far: the best known approximation algorithm achieves a roughly O(\sqrt m)-approximation, and in contrast, the best known hardness of approximation stands at $2$. Our main result is an approximation algorithm that achieves a $\tilde{O}(n^{\eps})$ approximation in time $n^{O(1/\eps)}$, for any $\eps=\Omega(\log\log n/\log n)$. In particular, we obtain poly-logarithmic approximation in quasi-polynomial time, and for every constant $\eps > 0$, we obtain an $O(n^{\eps})$-approximation in polynomial time. An interesting technical aspect of our algorithm is that we use as a building block a linear program whose integrality gap is $\Omega(\sqrt m)$. We bypass this obstacle by iteratively using the solutions produced by the LP to construct new instances with significantly smaller integrality gaps, eventually obtaining the desired approximation.

Joint work with Deeparnab Chakrabarty and Sanjeev Khanna