On the Value of Job Migration in Online Makespan Minimization

TL;DR

This paper investigates how limited job migration in online scheduling can improve makespan minimization, establishing optimal competitive ratios and migration bounds for deterministic algorithms on identical parallel machines.

Contribution

It introduces a deterministic online algorithm with proven competitive ratios using job migration, and provides matching lower bounds, advancing understanding of migration's impact.

Findings

Achieves an $oldsymbol{oldsymbol{ ext{α}_m}}$-competitive ratio with migration bounds.

Provides a lower bound showing no o(n) migration algorithm can do better.

Develops trade-off algorithms balancing competitiveness and migration count.

Abstract

Makespan minimization on identical parallel machines is a classical scheduling problem. We consider the online scenario where a sequence of $n$ jobs has to be scheduled non-preemptively on $m$ machines so as to minimize the maximum completion time of any job. The best competitive ratio that can be achieved by deterministic online algorithms is in the range $[1.88,1.9201]$. Currently no randomized online algorithm with a smaller competitiveness is known, for general $m$. In this paper we explore the power of job migration, i.e.\ an online scheduler is allowed to perform a limited number of job reassignments. Migration is a common technique used in theory and practice to balance load in parallel processing environments. As our main result we settle the performance that can be achieved by deterministic online algorithms. We develop an algorithm that is $\alpha_m$-competitive, for any $m\geq 2$, where $\alpha_m$ is the solution of a certain equation. For $m=2$, $\alpha_2 = 4/3$ and $\lim_{m\rightarrow \infty} \alpha_m = W_{-1}(-1/e^2)/(1+ W_{-1}(-1/e^2)) \approx 1.4659$. Here $W_{-1}$ is the lower branch of the Lambert $W$ function. For $m\geq 11$, the algorithm uses at most $7m$ migration operations. For smaller $m$, $8m$ to $10m$ operations may be performed. We complement this result by a matching lower bound: No online algorithm that uses $o(n)$ job migrations can achieve a competitive ratio smaller than $\alpha_m$. We finally trade performance for migrations. We give a family of algorithms that is $c$-competitive, for any $5/3\leq c \leq 2$. For $c= 5/3$, the strategy uses at most $4m$ job migrations. For $c=1.75$, at most $2.5m$ migrations are used.