Online NDP-constraint scheduling of jobs with delivery times or weights

This paper studies three online scheduling problems on the single-machine, denoted by (P1), (P2) and (P3), to minimize the maximum weighted completion time or the maximum delivery completion time with or without the NDP constraint. Here, "NDP" means that the available jobs cannot be delayed for processing when the machine is idle. Problems (P1) and (P2) aim to minimize the maximum weighted completion time, where (P1) has not the NDP constraint and (P2) has the NDP constraint. Problem (P3) aims to minimize the maximum delivery completion time under the NDP constraint. For problems (P1) and (P2), we study the restricted version in which the jobs have agreeable processing times and weights (i.e., if p(i) > p(j), then w(i) >= w(j)). For problem (P3), we study the restricted version in which the jobs have small delivery times (i.e., p(j) >= q(j)). We show that problem (P1) admits a best possible online algorithm with a competitive ratio of 1.618, problem (P2) has a lower bound 3/2 and admits a root 3-competitive online algorithm, and problem (P3) admits a best possible online algorithm with a competitive ratio of 4/3.