An improved bound for the price of anarchy for related machine scheduling

TL;DR

This work studies the price of anarchy for utilitarian scheduling on related machines under Smith's rule for the sum of completion times. It employs a primal–dual dual-fitting framework built on LP formulations to bound the PoA, achieving a tight bound of $\frac{3}{2}$ for the two-machine case and a general bound of $2-\frac{1}{2\cdot(2m-1)}$, with an improved $2-\frac{1}{2m}$ when speeds are divisible. The analysis uses structures like sub-chains and critical jobs to define feasible dual variables, and connects the equilibrium set to the Ibarra–Kim algorithm, yielding an approximation interpretation. Overall, the results tighten worst-case guarantees for decentralized scheduling and provide refined performance bounds for related algorithms in this domain.

Abstract

In this paper, we introduce an improved upper bound for the efficiency of Nash equilibria in utilitarian scheduling games on related machines. The machines have varying speeds and adhere to the Shortest Processing Time (SPT) policy as the global order for job processing. The goal of each job is to minimize its completion time, while the social objective is to minimize the sum of completion times. Our main result provides an upper bound of $2-\frac{1}{2\cdot(2m-1)}$ on the price of anarchy for the general case of $m$ machines. We improve this bound to 3/2 for the case of two machines, and to $2-\frac{1}{2\cdot m}$ for the general case of $m$ machines when the machines have divisible speeds.

Figures (1)

• Figure 1: The block structure of any optimal schedule when $m=2$.

Theorems & Definitions (5)

• theorem thmcountertheorem: Horowitz and Sahni horowitz1976exact
• lemma thmcounterlemma: Hoeksma and Uetz hoeksma2019price
• theorem thmcountertheorem: Heydenreich et al. heydenreich2007games, Immorlica et al immorlica
• theorem thmcountertheorem
• proof