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 $\mathbf{NP}$-hard for $m=3$.
We prove that for any fixed $\varepsilon$ and $m$, a Sherali-Adams / Lasserre lift of the time-index LP with a slightly super poly-logarithmic number of $r = (\log(n… Expand

It is shown that many problems can be $$\mathcal {NP}$$NP-hard when considering general precedence constraints, while they become polynomially solvable for particular precedence constraints.Expand

This dissertation presents three results which make modest progress towards understanding the power and limitations of the Sum-of-Squares Hierarchy; all three works use average-case problems as a lens for theSum-of theSquares algorithms, by enabling us to userandom matrix theory as a tool in the analysis.Expand

Lower bounds are provided for these hierarchies when applied over the configuration LP for the problem of scheduling identical machines to minimize the makespan.Expand

This work gives an algorithm with running time nearly linear in the input size that approximately recovers a planted sparse vector with up to constant relative sparsity in a random subspace of ℝn of dimension up to Ω(√n).Expand

In approximation algorithms, I am interested in the use of Linear Programming and Semidefinite Programming relaxations and hierarchies in approximating solutions to NP-hard problems.Expand