Tight Bounds for Online Vector Scheduling

TL;DR

This paper establishes tight bounds for online vector scheduling, revealing the optimal competitive ratios for identical and unrelated machines, and introduces novel techniques for analyzing complex multi-resource scheduling problems.

Contribution

It provides the first tight bounds for online vector scheduling on both identical and unrelated machines, using innovative lower bound constructions and potential function methods.

Findings

Optimal competitive ratio for identical machines is Θ(log d / log log d).

Optimal competitive ratio for unrelated machines is Θ(log m + log d).

New online lower bounds and algorithms matching these bounds.

Abstract

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional 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. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J of Comp. 2006), vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations. 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 mono-chromatic clique problem using a novel online coloring game and randomized coding scheme. -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.