An Optimal Control Framework for Online Job Scheduling with General Cost Functions

TL;DR

This paper develops an optimal control framework for online job scheduling with general cost functions, providing competitive algorithms for single and multiple machine settings and extending classical scheduling rules.

Contribution

It introduces a novel optimal control approach to design competitive online scheduling algorithms with general cost functions, including for unrelated machines.

Findings

Highest-density-first rule is optimal for certain single-machine cost functions.

Proposed a speed-augmented competitive algorithm for general nondecreasing cost functions.

Extended algorithms to multiple unrelated machines with convex cost functions.

Abstract

We consider the problem of online job scheduling on a single machine or multiple unrelated machines with general job/machine-dependent cost functions. In this model, each job $j$ has a processing requirement (length) $v_{ij}$ and arrives with a nonnegative nondecreasing cost function $g_{ij}(t)$ if it has been dispatched to machine $i$, and this information is revealed to the system upon arrival of job $j$ at time $r_j$. The goal is to dispatch the jobs to the machines in an online fashion and process them preemptively on the machines so as to minimize the generalized completion time $\sum_{j}g_{i(j)j}(C_j)$. Here $i(j)$ refers to the machine to which job $j$ is dispatched, and $C_j$ is the completion time of job $j$ on that machine. It is assumed that jobs cannot migrate between machines and that each machine can work on a single job at any time instance. In particular, we are interested in finding an online scheduling policy whose objective cost is competitive with respect to a slower optimal offline benchmark, i.e., the one that knows all the job specifications a priori and is slower than the online algorithm. We first show that for the case of a single machine and special cost functions $g_j(t)=w_jg(t)$, with nonnegative nondecreasing $g(t)$, the highest-density-first rule is optimal for the generalized fractional completion time. We then extend this result by giving a speed-augmented competitive algorithm for the general nondecreasing cost functions $g_j(t)$ by utilizing a novel optimal control framework. This approach provides a principled method for identifying dual variables in different settings of online job scheduling with general cost functions. Using this method, we also provide a speed-augmented competitive algorithm for multiple unrelated machines with convex functions $g_{ij}(t)$, where the competitive ratio depends on the curvature of cost functions $g_{ij}(t)$.