Improved Approximations for Unrelated Machine Scheduling

TL;DR

This paper improves approximation algorithms for unrelated machine scheduling problems, achieving better bounds for total weighted completion time and machine load norms through novel rounding schemes and algorithmic techniques.

Contribution

It introduces a new rounding scheme for scheduling that yields improved approximation ratios for two key unrelated machine scheduling problems.

Findings

Achieved a 1.45-approximation for total weighted completion time.

Developed a $ frac{2}{ ext{approx}}$-approximation for the $L_2$-norm of machine loads.

Enhanced previous approximation bounds with novel algorithmic insights.

Abstract

We revisit two well-studied scheduling problems in the unrelated machines setting where each job can have a different processing time on each machine. For minimizing total weighted completion time we give a 1.45-approximation, which improves upon the previous 1.488-approximation [Im and Shadloo SODA 2020]. The key technical ingredient in this improvement lies in a new rounding scheme that gives strong negative correlation with less restrictions. For minimizing $L_k$-norms of machine loads, inspired by [Kalaitzis et al. SODA 2017], we give better approximation algorithms. In particular we give a $\sqrt {4/3}$-approximation for the $L_2$-norm which improves upon the former $\sqrt 2$-approximations due to [Azar-Epstein STOC 2005] and [Kumar et al. JACM 2009].