Approximation algorithms for parallel machine scheduling with linear deterioration

This paper deals with a parallel machine scheduling problem. Different from fixed processing time assumption in the classical scheduling, a job's processing time is a simple linear increasing function of its starting time. The aim is makespan minimization, and our focus is on the case with an arbitrary number of parallel machines. We prove that LIST rule is (1+b_max)m-1/ m-approximation where m is the number of machines and b_max is the maximum deteriorating rate of job. We then propose one heuristic LDR (Largest deteriorating Rate first). The heuristic is proved by (1+b_min)m-1/m- approximation where b_min is the minimum deteriorating rate. We further show that this ratio is tight when m=2,3 and 4.