Lovász Local Lemma : Application to Santa Claus Problem

A similar problem is the Load-Balancing Problem (Min-Max Allocation), where there’s a 2approximation algorithm via LP. The Santa Claus Problem with general evaluation functions is difficult, and people are interested in special cases. In this lecture we’ll focus on the case where vij ∈ {0, vj}: each item j has a value vj , and each player either want the item (vij = vj) or do not want the item (vij = 0). Bansal and Sviridenko [BS06] showed that this problem can be reduced to a combinatorial question. In the combinatorial question (see Fig. 2), there are m players in m/l groups of size l. Each group has l− 1 big items (that has value k). Each player is interested in k small items (that has value 1). Each small item is interested by at most l players (the bipartite graph in Fig. 2 only shows players and small items). Use Si to denote the set of small items player i wants, the goal is to pick a subset of players P ⊆ [m] where exactly one player is picked from each group, and then for each i ∈ P , pick a subset Ti ⊆ Si, such that all Ti’s are pairwise disjoint, and the min size of the sets Ti is maximized.