Robust Scheduling on Uniform Machines -- New Results Using a Relaxed Approximation Guarantee

We consider the problem of scheduling $n$ jobs on $m$ uniform machines while minimizing the makespan ($Q||C_{\max}$) and maximizing the minimum completion time ($Q||C_{\min}$) in an online setting with migration of jobs. In this online setting, the jobs are inserted or deleted over time, and at each step, the goal is to compute a near-optimal solution while reassigning some jobs, such that the overall processing time of reassigned jobs, called migration, is bounded by some factor $\beta$ times the processing time of the job added or removed. We propose Efficient Polynomial Time Approximation Schemes (EPTASs) with an additional load error of $\mathcal{O}(\varepsilon p_{\max})$ for both problems, with constant amortized migration factor $\beta$, where $p_{\max}$ is the maximum processing time in the instance over all steps. As an intermediate step, we obtain Efficient Parameterized Approximation Schemes (EPASs) for both problems, $(1+\varepsilon)$-competitive algorithms parameterized by $p_{\max}$ and the number of different processing times $d$ in an instance, with $\beta$ bounded in a function of $p_{\max}$, $d$ and $\varepsilon$. This is the first result in the direction of a polynomial time approximation scheme in the field of online scheduling with bounded reassignment on uniform machines; before, such results were known only for the considered problems on identical machines. Crucial to our result is a division of the machines into large and small machines depending on the current approximate objective value, allowing for different approaches on either machine set, as well as a new way of rounding the instance that does not depend on the current objective value.