Approximation algorithms for the W-prize-collecting scheduling problem on a single machine with submodular rejection penalties

In this paper, we consider the W-prize-collecting scheduling problem on a single machine with submodular rejection penalties. In this problem, we are given one machine, n jobs and a value W. Every job has a processing time and a profit. Each job is either accepted and processed on the machine, or rejected and a rejection penalty is paid. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalties of the rejected jobs which is determined by a submodular function, provided that the total profit of the accepted jobs is at least W. Under the assumption that the submodular penalty function is polymatriod, we design a 2-approximation algorithm based on the primal-dual framework.