Semi-Online Scheduling on Two Identical Parallel Machines with Initial-Lookahead Information

This work investigates a new semi-online scheduling problem with lookahead. We focus on job scheduling on two identical parallel machines, where deterministic online algorithms only know the information of k initial jobs (i.e., the initial-lookahead information), while the following jobs still arrive one-by-one in an over-list fashion. We consider makespan minimization as the objective. The study aims at revealing the value of knowing k initial jobs, which are used to improve the competitive performance of those online algorithms without such initial-lookahead information. We provide the following findings: (1) For the scenario where the k initial jobs are all the largest jobs with length Δ, we prove that the classical LIST algorithm is optimal with competitive ratio (k + 3)/(k + 2); (2) For the scenario where the total length of these k (> 1) jobs is at least Δ, we show that any online algorithm has a competitive ratio at least 3/2, implying that the initial-lookahead knowledge is powerless since there exists a 3/2-competitive online algorithm without such information; (3) For the scenario where the total length of these k (> α) jobs is at least αΔ (α ≥ 2), we propose an online algorithm, named as LPT-LIST, with competitive ratio of (α + 2)/(α + 1), implying that the initial-lookahead information indeed helps to improve the competitiveness of those online algorithms lacking such information.