Online Job Scheduling with K Servers

In this paper, we investigate an online job scheduling problem with n jobs and k servers, where the accessibilities between the jobs and the servers are given as a bipartite graph. The scheduler is tasked with minimizing the regret, defined as the difference between the total flow time of the scheduler over T rounds and that of the best-fixed scheduling in hindsight. We propose an algorithm whose regret bounds are $O(n^2 \sqrt{T\ln (nk)})$ for general bipartite graphs, $O((n^2/k^{1/2}) \sqrt{T\ln (nk)})$ for the complete bipartite graphs, and $O((n^2/k) \sqrt{T \ln (nk)}$ for the disjoint star graphs, respectively. We also give a lower regret bound of $\Omega((n^2/k) \sqrt{T})$ for the disjoint star graphs, implying that our regret bounds are almost optimal.