Quasi-PTAS for Scheduling with Precedences using LP Hierarchies

TL;DR

This paper presents a quasi-polynomial time approximation scheme for scheduling jobs with precedence constraints on fixed number of machines, improving previous results by reducing the number of LP hierarchy rounds needed.

Contribution

It introduces a quasi-PTAS using a fixed number of Sherali-Adams hierarchy rounds for scheduling with precedence constraints, advancing the understanding of approximation algorithms in this domain.

Findings

Achieves a (1+ε)-approximation in quasi-polynomial time

Uses a fixed number of Sherali-Adams hierarchy rounds

Improves upon previous LP-based approximation methods

Abstract

A central problem in scheduling is to schedule $n$ unit size jobs with precedence constraints on $m$ identical machines so as to minimize the makespan. For $m=3$, it is not even known if the problem is NP-hard and this is one of the last open problems from the book of Garey and Johnson. We show that for fixed $m$ and $\epsilon$, $(\log n)^{O(1)}$ rounds of Sherali-Adams hierarchy applied to a natural LP of the problem provides a $(1+\epsilon)$-approximation algorithm running in quasi-polynomial time. This improves over the recent result of Levey and Rothvoss, who used $r=(\log n)^{O(\log \log n)}$ rounds of Sherali-Adams in order to get a $(1+\epsilon)$-approximation algorithm with a running time of $n^{O(r)}$.