Tight Bounds for Online Vector Scheduling

Abstract

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multidimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and $L_r$ norms of loads are among the most popular objectives considered. Furthermore, the server clusters are also often heterogeneous making the scheduling problem more challenging. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna in 2006, vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. The scalar problem, introduced by Graham in 1966, and its many variants (identical and unrelated machines, makespan and $L_r$ norm optimization, offline and online jobs, etc.) have been extensively studied over the last 50 years. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations---for all $L_r$ norms and in both the identical and unrelated machines settings. For an instance with $m$ machines and $d$ dimensions, our main results are: For identical machines, we show that the optimal competitive ratio is $\Theta(\log d / \log \log d)$ by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum monochromatic clique problem using a novel online coloring game and randomized coding scheme. Our techniques also extend to asymptotically tight upper and lower bounds for general $L_r$ norms. For unrelated machines, we show that the optimal competitive ratio is $\Theta(\log m + \log d)$ by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general $L_r$ norms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual $L_r$ objectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.