  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.
