Online Load and Graph Balancing for Random Order Inputs

TL;DR

The paper studies online makespan minimization for unrelated machines under random arrival order, addressing both lower and upper bounds. It proves an $\Omega(\sqrt{\log m})$ lower bound for graph balancing (even on trees) and presents an $O\left(\frac{\log m}{\log \log m}\right)$-competitive online algorithm for the general problem, using a two-phase restart and a softmax potential. This establishes a separation between random-order and adversarial-order models and provides a near-tight improvement in the random-order setting. The techniques combine fat-tree constructions for lower bounds with a phase-based, potential-driven algorithm to achieve tighter guarantees, with practical implications for scheduling on heterogeneous systems.

Abstract

Online load balancing for heterogeneous machines aims to minimize the makespan (maximum machine workload) by scheduling arriving jobs with varying sizes on different machines. In the adversarial setting, where an adversary chooses not only the collection of job sizes but also their arrival order, the problem is well-understood and the optimal competitive ratio is known to be $Θ(\log m)$ where $m$ is the number of machines. In the more realistic random arrival order model, the understanding is limited. Previously, the best lower bound on the competitive ratio was only $Ω(\log \log m)$. We significantly improve this bound by showing an $Ω( \sqrt {\log m})$ lower bound, even for the restricted case where each job has a unit size on two machines and infinite size on the others. On the positive side, we propose an $O(\log m/\log \log m)$-competitive algorithm, demonstrating that better performance is possible in the random arrival model.

Figures (2)

• Figure 1: Illustration of a lower bound instance giving $\Omega(\log m)$ lower bound in the adversarial arrival model. Edges arrive in the order of solid, dashed, and dotted, and they are wlog assumed to be oriented towards the root.
• Figure 2: Example of a lower bound instance for $D=2$. Consider the time when the dashed edge appears. Edges are dotted if they have already arrived, solid otherwise. For the dashed edge, no algorithm can distinguish which of its end points is the parent node.

Theorems & Definitions (22)

• Claim 1
• Claim 2
• proof
• Theorem 1
• Definition 1: Bad node
• Lemma 1
• proof
• Definition 2: Bad subtree
• Lemma 2
• proof