Quasi-PTAS for scheduling with precedences using LP hierarchies

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 , polylog(n) rounds of Sherali-Adams hierarchy applied to a natural LP of the problem provides a (1+)-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 + )-approximation algorithm with a running time of n^O(r^)..