In this paper, we consider the Santa Claus problem in the CONGEST model. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. Notably, each gift can have a different value. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we obtain the first results the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\hat Θ(\sqrt n+D)$ rounds, where our $\widetildeΩ(\sqrt n+D)$-round lower bound is even stronger and holds for any approximation.

翻译：本文研究了CONGEST模型下的圣诞老人问题。该NP难问题可建模为儿童与礼物构成的二分图，其中边表示儿童渴望某件礼物。值得注意的是，每件礼物具有不同价值。目标是将礼物分配给儿童，使得最不快乐的儿童尽可能快乐。尽管这是串行设置中研究充分的问题，我们首次获得了分布式设置下的研究成果。具体而言，我们证明了计算$\mathcal{O}(\log n/\log \log n)$近似解的复杂度为$\hat Θ(\sqrt n+D)$轮次，其中$\widetildeΩ(\sqrt n+D)$轮的下界更为严格，且适用于任意近似比。