A Combinatorial 2-Approximation Algorithm for the Parallel-Machine Scheduling with Release Times and Submodular Penalties

Abstract

In this paper, we consider parallel-machine scheduling with release times and submodular penalties (P|rj,reject|Cmax+π(R)), in which each job can be accepted and processed on one of m identical parallel machines or rejected, but a penalty must paid if a job is rejected. Each job has a release time and a processing time, and the job can not be processed before its release time. The objective of P|rj,reject|Cmax+π(R) is to minimize the makespan of the accepted jobs plus the penalty of the rejected jobs, where the penalty is determined by a submodular function. This problem generalizes a multiprocessor scheduling problem with rejection, the parallel-machine scheduling with submodular penalties, and the single machine scheduling problem with release dates and submodular rejection penalties. In this paper, inspired by the primal-dual method, we present a combinatorial 2-approximation algorithm to P|rj,reject|Cmax+π(R). This ratio coincides with the best known ratio for the parallel-machine scheduling with submodular penalties and the single machine scheduling problem with release dates and submodular rejection penalties.

A Combinatorial Approximation Algorithm for the Vector Scheduling with Submodular Penalties on Parallel Machines

“…Tey presented a combinatorial (2 − 1/m)-approximation algorithm based on the list scheduling (LS) algorithm and a greedy method. Wang and Liu [18] focused on parallel-machine scheduling with release times and submodular penalties, proposing a combinatorial 2-approximation algorithm. Zhang et al [19] tackled the precedence-constrained scheduling problem with submodular rejection on parallel machines.…”