Approximation algorithms for the multiprocessor scheduling with submodular penalties

Abstract

In this paper, we consider multiprocessor scheduling with submodular penalties to extend multiprocessor scheduling with rejection to submodular function. An instance of the problem is given by n jobs and m machines with each job having a certain processing time on a machine. We aim to find a subset R of rejected jobs, and assign each of other jobs to one of the m machines. The objective is to minimize the sum of the makespan of the m machines and the rejection penalty R, where the rejection penalty is determined by a submodular function. For this problem, we design a non-combinatorial Lovasz rounding algorithm that achieves a worst-case guarantee of $$\frac{3+\sqrt{5}}{2}$$ . Then, we consider a special case of this problem in which all the machines are identical, i.e. each job has the same processing time on any machine, and we design a combinatorial $$(2-\frac{1}{m})$$ -approximation algorithm based on the greedy method and list scheduling (LS) algorithm.