Improved Scheduling with a Shared Resource

TL;DR

This paper presents new algorithms for shared-resource scheduling, achieving optimal competitive ratios for makespan minimization and improved approximation factors for total completion time, in both online and fractional settings.

Contribution

It introduces an optimal online algorithm for makespan minimization and a better approximation algorithm for total completion time using CLP and geometric analysis.

Findings

Optimal $e/(e-1)$-competitive algorithm for online makespan minimization.

A $(3/2+ ext{epsilon})$-approximation for total completion time.

Improved approximation factor from 2 to 1.5+epsilon for fractional total completion time.

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 $r_j\in[0,1]$, 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 $\mathcal{O}(n\cdot \log n)$. 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 $2$.