Narrow Results

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 $\mathrm{\Delta }$, 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 $\mathrm{\Delta }$, 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 $k(>\alpha )$ jobs is at least $\alpha \mathrm{\Delta }$ ($\alpha \ge 2$), we propose an online algorithm, named as LPT-LIST, with competitive ratio of $(\alpha +2)/(\alpha +1)$, implying that the initial-lookahead information indeed helps to improve the competitiveness of those online algorithms lacking such information.