Improved Scheduling with a Shared Resource

Abstract

We consider the following shared-resource scheduling problem: Given a set of jobs J, for each $j\in J$ we must schedule a job-specific processing volume of $v_j>0$. A total resource of 1 is available at any time. Jobs have a resource requirement $r_j\in \left[ 0,1 \right]$, and the resources assigned to them may vary over time. However, assigning them less will cause a proportional slowdown.

We consider two settings. In the first, we seek to minimize the makespan in an online setting: The resource assignment of a job must be fixed before the next job arrives. Here we give an optimal $e/(e-1)$-competitive algorithm with runtime $\text {O}({n\text {\,\,} \log \, n})$. In the second, we aim to minimize the total completion time. We use a continuous linear programming (CLP) formulation for the fractional total completion time and combine it with a previously known dominance property from malleable job scheduling to obtain a lower bound on the total completion time. We extract structural properties by considering a geometrical representation of a CLP’s primal-dual pair. We combine the CLP schedule with a greedy schedule to obtain a $(3/2+\varepsilon )$-approximation for this setting. This improves upon the so far best-known approximation factor of 2.

Full Version (Preprint): https://arxiv.org/abs/2310.05732.