Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations

TL;DR

This paper develops improved approximation algorithms for scheduling problems aimed at minimizing total weighted completion time, utilizing novel time-indexed linear programming relaxations that outperform previous methods and simplify existing approaches.

Contribution

It introduces a natural, simpler time-indexed LP relaxation for scheduling problems, achieving near state-of-the-art approximation ratios and offering new insights.

Findings

Improved approximation algorithms for scheduling problems.

Time-indexed LP relaxations outperform previous methods.

Achieved a 1.5 - c approximation for unrelated machines scheduling.

Abstract

We study approximation algorithms for scheduling problems with the objective of minimizing total weighted completion time, under identical and related machine models with job precedence constraints. We give algorithms that improve upon many previous 15 to 20-year-old state-of-art results. A major theme in these results is the use of time-indexed linear programming relaxations. These are natural relaxations for their respective problems, but surprisingly are not studied in the literature. We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines. The recent breakthrough result of [Bansal-Srinivasan-Svensson, STOC 2016] gave a $(1.5-c)$-approximation for the problem, based on some lift-and-project SDP relaxation. Our main result is that a $(1.5 - c)$-approximation can also be achieved using a natural and considerably simpler time-indexed LP relaxation for the problem. We hope this relaxation can provide new insights into the problem.