Robust Polynomial-Time Approximation Schemes for Parallel Machine Scheduling with Job Arrivals and Departures

Scheduling a set of n jobs on m identical parallel machines so as to minimize the makespan or maximize the minimum machine load are two of the most important and fundamental scheduling problems studied in the literature. We consider the general online scenario where jobs are consecutively added to and/or deleted from an instance. The goal is to maintain a near-optimal assignment of the current set of jobs to the m machines. This goal is essentially doomed to failure unless, upon arrival or departure of a job, we allow reassigning some other jobs. Considering that the reassignment of a job induces a cost proportional to its size, the total cost for reassigning jobs must preferably be bounded by a constant r times the total size of added or deleted jobs. The value r is called the reassignment factor of the solution and it is a measure of our willingness to adapt the solution over time.

Our main result is that, for any ε > 0, it is possible to achieve (1 + ε)-competitive solutions with constant reassignment factor r(ε). For the minimum makespan problem this is the first improvement on the (2 + ε)-competitive algorithm by Andrews et al. (1999) [Andrews M, Goemans M, Zhang L (1999) Improved bounds for on-line load balancing. Algorithmica 23(4):278–301]. Crucial to our algorithm is a new insight into the structure of robust, almost optimal schedules.