Santa Claus meets Makespan and Matroids: Algorithms and Reductions
\'Etienne Bamas, Alexander Lindermayr, Nicole Megow, Lars Rohwedder,, Jens Schl\"oter

TL;DR
This paper explores the deep connection between makespan minimization and the Santa Claus problem, establishing reductions, equivalences, and an algorithm for a matroid-based variant, advancing understanding of their approximability.
Contribution
It introduces formal reductions linking the two problems, especially for two-value and matroid-constrained cases, and presents a new algorithm for a generalized Santa Claus problem.
Findings
Reduces Santa Claus to makespan minimization, implying similar approximation bounds.
Shows equivalence of the two problems for instances with only two input values.
Provides an algorithm for a matroid-constrained Santa Claus variant.
Abstract
In this paper we study the relation of two fundamental problems in scheduling and fair allocation: makespan minimization on unrelated parallel machines and max-min fair allocation, also known as the Santa Claus problem. For both of these problems the best approximation factor is a notorious open question; more precisely, whether there is a better-than-2 approximation for the former problem and whether there is a constant approximation for the latter. While the two problems are intuitively related and history has shown that techniques can often be transferred between them, no formal reductions are known. We first show that an affirmative answer to the open question for makespan minimization implies the same for the Santa Claus problem by reducing the latter problem to the former. We also prove that for problem instances with only two input values both questions are equivalent. We…
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