Comparing the minimum completion times of two longest-first scheduling-heuristics

For the problem of non-preemptively scheduling n independent jobs on m identical parallel machines so that the minimum (or earliest) machine completion time is maximized, we compare two well-known longest-first heuristics the LPT(longest processing time) and the RLPT-heuristic (restricted LPT). We prove that the minimum completion time of the LPT-schedule is at least as long as the minimum completion time of the RLPT-schedule. Furthermore, we show that the minimum completion time of the RLPT-heuristic always remains within a factor of 1/m of the optimal minimum completion time. The paper finishes with a conjecture on the probabilistic behavior of the RLPT-heuristic compared to the LPT-heuristic in case of two machines.