Single machine Pareto scheduling with positional deadlines and agreeable release and processing times

This paper studies the problem of scheduling $n$ jobs on a single machine to minimize total completion time and maximum cost, simultaneously. Each job is associated with a positional deadline that indicates the largest ordinal number of this job in any feasible schedule. The jobs have agreeable release and processing times, meaning that jobs with larger release times also have larger processing times. The agreeability assumption is reasonable since both the single-criterion problems (without positional deadline constraints) of minimizing total completion time and maximum lateness on a single machine with arbitrary release and processing times are strongly NP-hard. An $O(n^3)$-time Pareto optimal algorithm is presented. The previously known algorithms only solve two special cases of the agreeability assumption: either the case of equal release times in $O(n^4)$ time, or the case of equal processing times (without positional deadline constraints) in $O(n^3)$ time.