Allocating Goods to Maximize Fairness

Опубликовано 17 августа 2016, 1:39

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 the items it receives, and we seek to maximize the minimum utility of any agent. While this problem has received much attention, its approximability has not been well-understood thus far: the best previously known approximation algorithm achieved a roughly O(\sqrt m)-approximation, and in contrast, the best known hardness of approximation stands at factor $2$. We present 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. Joint work with Deeparnab Chakrabarty and Sanjeev Khanna