The Longest Processing Time rule for identical parallel machines revisited

We consider the Pm || Cmax scheduling problem where the goal is to schedule n jobs on m identical parallel machines (m = 3 and from 7/6 to 9/8 for m = 2. We use linear programming to analyze the approximation ratio of our approach. This performance analysis can be seen as a valid alternative to formal proofs based on analytical derivation. Also, we derive from the proposed approach an O(n log n) time complexity heuristic. The heuristic splits the sorted job set in tuples of m consecutive jobs (1,...,m; m+1,...,2m; etc.) and sorts the tuples in non-increasing order of the difference (slack) between largest job and smallest job in the tuple. Then, given this new ordering of the job set, list scheduling is applied. This approach strongly outperforms LPT on benchmark literature instances and is competitive with more involved approaches such as COMBINE and LDM