Allocating Indivisible Goods

Ivona Bezakova; Varsha Dani. 14 December, 2004.
Communicated by Lance Fortnow.


Given k players, m indivisible goods and a valuation function on the set of the goods for every player, Lipton, Markakis, Mossel and Saberi propose the Max-Min Fairness problem: partition the goods between the players such that the minimum value achieved by every player is maximized. We show that for two players and additive valuation functions, the expected minimum of the (randomized) cut-andchoose mechanism is a 1/2-approximation of the optimum. To complement this result we show that no truthful mechanism can compute the exact optimum.

We also consider the case when the algorithm knows the true additive valuations of every player. We present a simple 1/(m − k + 1) approximation algorithm which allocates to every player at least 1/k fraction of the value of all but the k − 1 heaviest items. We also give an algorithm which guarantees every player at least the fractional optimum of the corresponding linear relaxation, minus the largest item. This approximation algorithm has additive error at most the value of the largest item. We conclude with a 1/2 + e factor hardness of approximation result unless P = NP.

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