Online Load Balancing for Related Machines

In the load balancing (or job scheduling) problem, introduced by Graham in the 1960s (SIAM J. of Appl. Math. 1966, 1969), jobs arriving online have to be assigned to machines so to minimize an objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm and arbitrary lq-norms of machine loads. Recent literature (e.g., Azar et al., STOC 2013; Im et al., FOCS 2015) has further expanded the scope of this problem to vector loads, to capture jobs with multi-dimensional resource requirements in applications such as data centers. In this paper, we completely resolve the job scheduling problem for both scalar and vector jobs on related machines, i.e., where each machine has a given speed and the time taken to process a job is inversely proportional to the speed of the machine it is assigned on. We show the following results: • Scalar scheduling. We give a constant competitive algorithm for optimizing any lq-norm for (scalar) scheduling on related machines. The only previously known result was for the makespan norm. • Vector scheduling. There are two natural variants for vector scheduling, depending on whether the speed of a machine is dimension-dependent or not. We show a sharp contrast between these two variants, proving that they are respectively equivalent to unrelatedmachines and identical machines for the makespan norm. We also extend these results to arbitrary lq-norms of the machine loads. No previous results were known for vector scheduling of related machines. A key component of our algorithms is a new tool that we call machine smoothing, where we replace an arbitrary instance with a smoothed instance of the problem. The structural properties of the smoothed instance make it much simpler to argue about various norms of machine loads. We hope that this generic technique will find more applications in other scheduling problems as well. Email: [email protected]. Supported in part by NSF grants CCF-1409130 and CCF-1617653. Email: [email protected]. Supported in part by NSF awards CCF-1527084 and CCF-1535972. Email: [email protected]. Supported in part by NSF awards CCF-1527084 and CCF-1535972. Email: [email protected]. Supported in part by NSF grant CCF-1409130 and CCF-1617653.