Online load balancing on related machines

In this paper, we consider the problem of assigning jobs online to machines with non-uniform speeds (also called related machines) so to optimize a given norm of the machine loads. A long line of work, starting with the seminal work of Graham in the 1960s, has led to tight competitive ratios for all ℓq norms for two scenarios: the special case of identical machines (uniform machine speeds) and the more general setting of unrelated machines (jobs have arbitrary processing times on machines). For non-uniform machine speeds, however, the only known result was a constant competitive competitive ratio for the makespan (ℓ∞) norm, via the so-called slowest-fit algorithm (Aspnes, Azar, Fiat, Plotkin, and Waarts, JACM ’97). Our first result in this paper is to obtain the first constant-competitive algorithm for scheduling on related machines for any arbitrary ℓq norm. Recent literature has further expanded the scope of this problem to vector scheduling, to capture multi-dimensional resource requirements in applications such as data centers. As in the scalar case, tight bounds are known for vector scheduling on identical and unrelated machines. Our second set of results is to give tight competitive ratios for vector scheduling on related machines for the makespan and all ℓq norms. No previous bounds were known, even for the makespan norm, for related machines. We employ a convex relaxation of the ℓq-norm objective and use a continuous greedy algorithm to solve this convex program online. To round the fractional solution, we then use a novel restructuring of the instance that we call machine smoothing. This is a generic tool that reduces a problem on related machines to a set of problem instances on identical machines, and we hope it will be useful in other settings with non-uniform machine speeds as well.