Improved Lower Bounds for Truthful Scheduling

TL;DR

This paper advances the theoretical understanding of truthful scheduling mechanisms by establishing new, higher lower bounds on the approximation ratio, demonstrating that no mechanism can achieve better than these bounds even with small problem sizes.

Contribution

The paper introduces improved lower bounds for truthful scheduling mechanisms, including a bound of 3 - δ for any δ > 0, and simpler proofs for bounds with small numbers of machines and jobs.

Findings

Lower bound of 2.2055 with 3 machines and 3 jobs

Lower bound of 1+√2 with 3 machines and 4 jobs

Lower bound of 2 for 2 machines and 2 jobs

Abstract

The problem of scheduling unrelated machines by a truthful mechanism to minimize the makespan was introduced in the seminal "Algorithmic Mechanism Design" paper by Nisan and Ronen. Nisan and Ronen showed that there is a truthful mechanism that provides an approximation ratio of $\min(m,n)$, where $n$ is the number of machines and $m$ is the number of jobs. They also proved that no truthful mechanism can provide an approximation ratio better than $2$. Since then, the lower bound was improved to $1 +\sqrt 2 \approx 2.41$ by Christodoulou, Kotsoupias, and Vidali, and then to $1+\phi\approx 2.618$ by Kotsoupias and Vidali. Very recently, the lower bound was improved to $2.755$ by Giannakopoulos, Hammerl, and Pocas. In this paper we further improve the bound to $3-\delta$, for every constant $\delta>0$. Note that a gap between the upper bound and the lower bounds exists even when the number of machines and jobs is very small. In particular, the known $1+\sqrt{2}$ lower bound requires at least $3$ machines and $5$ jobs. In contrast, we show a lower bound of $2.2055$ that uses only $3$ machines and $3$ jobs and a lower bound of $1+\sqrt 2$ that uses only $3$ machines and $4$ jobs. For the case of two machines and two jobs we show a lower bound of $2$. Similar bounds for two machines and two jobs were known before but only via complex proofs that characterized all truthful mechanisms that provide a finite approximation ratio in this setting, whereas our new proof uses a simple and direct approach.