On allocations that maximize fairness

Abstract

We consider a problem known as the restricted assignment version of the max-min allocation problem with indivisible goods. There are n items of various nonnegative values and m players. Every player is interested only in some of the items and has zero value for the other items. One has to distribute the items among the players in a way that maximizes a certain notion of fairness, namely, maximizes the minimum of the sum of values of items given to any player. Bansal and Sviridenko [STOC 2006] describe a linear programming relaxation for this problem, and present a rounding technique that recovers an allocation of value at least Ω(log log log m / log log m) of the optimum. We show that the value of this LP relaxation in fact approximates the optimum value to within a constant factor. Our proof is not constructive and does not by itself provide an efficient algorithm for finding an allocation that is within constant factors of optimal. 1