LAZY LOCAL SEARCH MEETS MACHINE SCHEDULING.

We study the restricted case of Scheduling on Unrelated Parallel Machines. In this problem, we are given a set of jobs J with processing times pj and each job may be scheduled only on some subset of machines Sj⊆ M. The goal is to find an assignment of jobs to machines to minimize the time within which all jobs can be processed. In a seminal paper, Lenstra, Shmoys, and Tardos [Proceedings of the 28th Annual IEEE Conference on Foundations of Computer Science, 1987, pp. 217--224] designed an elegant 2-approximation for the problem. The question of whether approximation algorithms with better guarantees exist for this classic scheduling problem has since remained a source of mystery. In recent years, with the improvement of our understanding of configuration linear programs, it now appears an attainable goal to design such an algorithm. Our main contribution in this paper is to make progress towards this goal. For the so-called (1, ε)-case, when the processing times of jobs are either 1 or ε ∊ (0, 1), we design an approximation algorithm whose guarantee tends to 1+√3/2 ≈ 1.866025 as ε tends to zero. This improves on the 2 ε0 guarantee recently obtained by Chakrabarty, Khanna, and Li [Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2015, pp. 1087--1101] for some constant ε0 > 0. Further, the ideas presented in this paper do not crucially depend on the fact that the instances contain exactly two different job sizes; i.e., they extend also to families of instances more general than the (1, ε)-case. In a certain precise sense, we fall slightly short of improving on the 30-year-old classic algorithm of Lenstra, Shmoys, and Tardos [Proceedings of the 28th Annual IEEE Conference on Foundations of Computer Science, 1987, pp. 217--224] for the restricted case. [ABSTRACT FROM AUTHOR]