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

TL;DR

This paper develops new efficient approximation schemes for online scheduling on uniform machines, balancing near-optimal solutions with limited job reassignments, extending prior results from identical to uniform machines.

Contribution

It introduces the first polynomial time approximation schemes for online scheduling with bounded reassignment on uniform machines, using novel rounding and machine division techniques.

Findings

EPTAS with load error $ ext{O}( extvarepsilon p_{ ext{max}})$ and constant migration factor.

EPAS algorithms parameterized by maximum processing time and number of distinct processing times.

First polynomial time approximation schemes for online scheduling with bounded reassignment on uniform machines.

Abstract

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.