In a classical problem in scheduling, one has n unit size jobs with a precedence order and the goal is to find a schedule of those jobs on m identical machines as to minimize the makespan. It is one of the remaining four open problems from the book of Garey & Johnson whether or not this problem is NP-hard for m=3.
We prove that for any fixed epsilon > 0 and m, an LP-hierarchy lift of the time-index LP with a slightly super poly-logarithmic number of rounds provides a (1 + epsilon)-approximation. For example Sherali-Adams suffices as hierarchy. This implies an algorithm that yields a (1+epsilon)-approximation in almost quasi-polynomial time. The previously best approximation algorithms guarantee only a 2-approximation for large m.
This is joint work with Elaine Levey.