Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines

TL;DR

This paper presents a novel approximation algorithm for minimizing weighted completion times on unrelated machines, improving the approximation ratio below 1.5 using a new SDP relaxation and bipartite rounding technique.

Contribution

It introduces a new lift-and-project SDP relaxation and a bipartite-rounding method, achieving an improved approximation ratio for the scheduling problem.

Findings

Achieves a (3/2 - c)-approximation ratio, improving upon the previous 3/2 bound.

Develops a new SDP relaxation with better integrality gap.

Proposes a bipartite-rounding procedure with strong negative correlation properties.

Abstract

We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a $(3/2-c)$-approximation algorithm for some fixed $c>0$, improving upon the long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM, 2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce a new lift-and-project based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of $3/2$. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.