arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2007.04362cs.GT
收藏

改进的真调度下界

Improved Lower Bounds for Truthful Scheduling

Shahar Dobzinski, Ariel Shaulker

摘要:

在Nisan和Ronen的开创性论文“机器人机制设计“中,介绍了通过真实机制调度不相关机器以最小化最大完工时间的问题。Nisan和Ronen表明,存在一种真实机制,它提供了近似比率$\min(m,n)$,其中$n$是机器的数量,$m$是作业的数量。他们还证明,没有任何真实的机制可以提供比$2$更好的近似比。此后,Christodoulou、Kotsoupias和Vidali将下界改进为$1 +\sqrt 2 \approx 2.41$,然后Kotsoupias和Vidali将其改进为$1+ϕ\approx 2.618$。最近,Giannakopoulos、Hammerl和Pocas将下界改进为$2.755$。本文对每个常数$δ>0$,进一步改进了对$3-δ$的界.请注意,即使机器和作业的数量很小,上限和下限之间也存在差距。特别地,已知的$1+\sqrt{2}$下限至少需要$3$机器和$5$作业。相比之下,我们显示了一个下界的$2.2055$,只使用$3$机器和$3$工作和下界的$1+\sqrt 2$,只使用$3$机器和$4$工作。对于两台机器和两个工作的情况下,我们展示了$2$的下界。两台机器和两个工作的类似界限以前是已知的,但只有通过复杂的证明,其特征在于所有真实的机制,在这种情况下提供一个有限的近似比,而我们的新证明使用一个简单而直接的方法。

英文摘要:

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 $1+ϕ\approx 2.618$ 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 $3-δ$, for every constant $δ>0$. 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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

↑