A New Lower Bound for Deterministic Truthful Scheduling

TL;DR

This paper improves the known lower bounds for the approximation ratio of deterministic truthful scheduling mechanisms, showing that the ratio can be at least 2.755 even with many machines, advancing understanding in algorithmic mechanism design.

Contribution

It provides the first significant improvement in over a decade on lower bounds for deterministic truthful scheduling, reducing the gap to the optimal approximation ratio.

Findings

Lower bound of 2.618 achieved with 4 machines

Lower bound of 2.711 with 5 machines

Lower bound of 2.755 for infinitely many machines

Abstract

We study the problem of truthfully scheduling $m$ tasks to $n$ selfish unrelated machines, under the objective of makespan minimization, as was introduced in the seminal work of Nisan and Ronen [STOC'99]. Closing the current gap of $[2.618,n]$ on the approximation ratio of deterministic truthful mechanisms is a notorious open problem in the field of algorithmic mechanism design. We provide the first such improvement in more than a decade, since the lower bounds of $2.414$ (for $n=3$) and $2.618$ (for $n\to\infty$) by Christodoulou et al. [SODA'07] and Koutsoupias and Vidali [MFCS'07], respectively. More specifically, we show that the currently best lower bound of $2.618$ can be achieved even for just $n=4$ machines; for $n=5$ we already get the first improvement, namely $2.711$; and allowing the number of machines to grow arbitrarily large we can get a lower bound of $2.755$.