Efficient Approximation Algorithms for Scheduling Unrelated Parallel Machines

Scheduling n independent jobs on m Unrelated Parallel Machines (SUM) is the problem of assigning n jobs j = 1, .., n to m machines i = 1, .., m so that each job is processed without interruption on one of the machines, and at any time, every machine processes at most one job. The objective is to minimize the makespan of the schedule. SUM is an NP-hard problem even when the number of machines is defined to be m = 2. In this work, efficient approximation algorithms for SUM, when the number of machines is an arbitrary constant, are presented. The results hold for SUM and several extensions of SUM, like the Generalized Assignment Problem (GAP). The approximation algorithms can achieve any given approximation ratio 2 (approximation schemes) and admit efficient parallelization. The core of the algorithms is a new rounding procedure, the so-called Combinatorial Randomized Rounding (CRR) technique. Furthermore, an interesting property of combinatorial optimization problems and a new Chernoff-like bound, are introduced.