An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods

In this paper, we give the first approximation algorithm for the problem of max-min fair allocation of indivisible goods. An instance of this problem consists of a set of $k$ people and $m$ indivisible goods. Each person has a known linear utility function over the set of goods which might be different from the utility functions of other people. The goal is to distribute the goods among the people and maximize the minimum utility received by them. The approximation ratio of our algorithm is $\Omega(\frac{1}{\sqrt{k}\log^{3}k})$. As a crucial part of our algorithm, we design and analyze an iterative method for rounding a fractional matching on a tree which might be of independent interest. We also provide better bounds when we are allowed to exclude a small fraction of the people from the problem.

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