The Santa Claus problem

We consider the following problem: The Santa Claus has n presents that he wants to distribute among m kids. Each kid has an arbitrary value for each present. Let p<sub>ij</sub> be the value that kid i has for present j. The Santa's goal is to distribute presents in such a way that the least lucky kid is as happy as possible, i.e he tries to maximize min<sub>i=1,...,m</sub> sum<sub>j ∈ S<sub>i</sub></sub> p<sub>ij</sub> where S<sub>i</sub> is a set of presents received by the i-th kid.Our main result is an O(log log m/log log log m) approximation algorithm for the restricted assignment case of the problem when p<sub>ij</sub> ∈ p<sub>j</sub>,0 (i.e. when present j has either value p<sub>j</sub> or 0 for each kid). Our algorithm is based on rounding a certain natural exponentially large linear programming relaxation usually referred to as the configuration LP. We also show that the configuration LP has an integrality gap of Ω(m<sup>1/2</sup>) in the general case, when p<sub>ij</sub> can be arbitrary.

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