Online Makespan Minimization: Beat LPT by Dynamic Locking
Online makespan minimization is a fundamental problem in scheduling. In this paper, we investigate its over-time formulation, where each job has a release time and a processing time. A job becomes known only at its release time and must be scheduled on a machine thereafter. The Longest Processing Time First (LPT) algorithm, established by Chen and Vestjens (1997), achieves a competitive ratio of $1.5$. For the special case of two machines, Noga and Seiden introduced the SLEEPY algorithm, which achieves a tight competitive ratio of $1.382$. However, for $m \geq 3$, no known algorithm has convincingly surpassed the long-standing $1.5$ barrier. We propose a natural generalization of SLEEPY and show this simple approach can beat the $1.5$ barrier and achieve $1.482$-competitive when $m=3$. However, when $m$ becomes large, we prove this simple generalization fails to beat $1.5$. Meanwhile, we introduce a novel technique called dynamic locking to overcome this new challenge. As a result, we achieve a competitive ratio of $1.5-\frac{1}{O(m^2)}$, which beats the LPT algorithm ($1.5$-competitive) for every constant $m$.