Distributed Santa Claus via Global Rounding

2026-04-30

AI Summary

This work addresses the Santa Claus problem in the CONGEST model, where heterogeneous gifts must be distributed among children in a network to maximize the minimum satisfaction. It presents the first theoretical breakthrough for this problem in a distributed setting by modeling it via bipartite graphs and employing a global rounding technique to devise an algorithm achieving an approximation ratio of \(O(\log n / \log \log n)\) within \(\tilde{\Theta}(\sqrt{n} + D)\) communication rounds, where \(n\) is the number of nodes and \(D\) the network diameter. Furthermore, the paper establishes a matching lower bound by proving that any approximation algorithm requires \(\tilde{\Omega}(\sqrt{n} + D)\) rounds, thereby tightly characterizing the inherent communication complexity of the problem under bandwidth constraints.

Abstract

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.

Problem

• Santa Claus problem
• distributed algorithms
• CONGEST model
• approximation
• bipartite graph

Innovation

• Distributed Algorithms
• Santa Claus Problem
• CONGEST Model
• Approximation Algorithms
• Round Complexity