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On Allocating Goods to Maximize Fairness
, 2009
"... Given a set A of m agents and a set I of n items, where agent A ∈ A has utility uA,i for item i ∈ I, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of the utilities for items it receives, and we seek to maximize the minimum utility of a ..."
Abstract

Cited by 11 (0 self)
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Given a set A of m agents and a set I of n items, where agent A ∈ A has utility uA,i for item i ∈ I, our goal is to allocate items to agents to maximize fairness. Specifically, the utility of an agent is the sum of the 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 wellunderstood thus far: the best known approximation algorithm achieves an Õ(√m)approximation, and in contrast, the best known hardness of approximation stands at 2. Our main result is an approximation algorithm that achieves an Õ(nɛ) approximation for any ɛ = Ω(log log n / log n) in time nO(1/ɛ). In particular, we obtain polylogarithmic approximation in quasipolynomial time, and for every constant ɛ> 0, we obtain Õ(nɛ)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 Ω ( √ 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. We also investigate the special case of the problem, where every item has a nonzero utility for at most two agents. We show that even in this restricted setting the problem is hard to approximate upto any factor better than 2, and show a factor (2 + ɛ)approximation algorithm running in time poly(n, 1/ɛ) for any ɛ> 0. This special case can be cast as a graph edge orientation problem, and our algorithm can be viewed as a generalization of Eulerian orientations to weighted graphs. 1
Santa Claus Meets Hypergraph Matchings
, 2008
"... We consider the problem of maxmin fair allocation of indivisible goods. Our focus will be on the restricted version of the problem in which there are m items, each of which associated with a nonnegative value. There are also n players and each player is only interested in some of the items. The go ..."
Abstract

Cited by 8 (1 self)
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We consider the problem of maxmin fair allocation of indivisible goods. Our focus will be on the restricted version of the problem in which there are m items, each of which associated with a nonnegative value. There are also n players and each player is only interested in some of the items. The goal is to distribute the items between the players such that the least happy person is as happy as possible, i.e. one wants to maximize the minimum of the sum of the values of the items given to any player. This problem is also known as the Santa Claus problem [3]. Feige [9] proves that the integrality gap of a certain configuration LP, described by Bansal and Sviridenko [3], is bounded from below by some (unspecified) constant. This gives an efficient way to estimate the optimum value of the problem within a constant factor. However, the proof in [9] is nonconstructive: it uses the Lovasz local lemma and does not provide a polynomial time algorithm for finding an allocation. In this paper, we take a different approach to this problem, based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is bounded by 1 5. Our proof is nonconstructive in the following sense: it does provide a local search algorithm which finds the corresponding allocation, but this algorithm is not known to converge to a local optimum in a polynomial number of steps. 1