Minimizing Makespan in Sublinear Time via Weighted Random Sampling

We consider the classical makespan minimization scheduling problem where $n$ jobs must be scheduled on $m$ identical machines. Using weighted random sampling, we developed two sublinear time approximation schemes: one for the case where $n$ is known and the other for the case where $n$ is unknown. Both algorithms not only give a $(1+3\epsilon)$-approximation to the optimal makespan but also generate a sketch schedule. Our first algorithm, which targets the case where $n$ is known and draws samples in a single round under weighted random sampling, has a running time of $\tilde{O}(\tfrac{m^5}{\epsilon^4} \sqrt{n}+A(\ceiling{m\over \epsilon}, {\epsilon} ))$, where $A(\mathcal{N}, \alpha)$ is the time complexity of any $(1+\alpha)$-approximation scheme for the makespan minimization of $\mathcal{N}$ jobs. The second algorithm addresses the case where $n$ is unknown. It uses adaptive weighted random sampling, %\textit{that is}, it draws samples in several rounds, adjusting the number of samples after each round, and runs in sublinear time $\tilde{O}\left( \tfrac{m^5} {\epsilon^4} \sqrt{n} + A(\ceiling{m\over \epsilon}, {\epsilon} )\right)$. We also provide an implementation that generates a weighted random sample using $O(\log n)$ uniform random samples.