# The Longest Processing Time rule for identical parallel machines revisited

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## Abstract

We consider the \(P_m || C_{\max }\) scheduling problem where the goal is to schedule *n* jobs on *m* identical parallel machines \((m < n)\) to minimize makespan. We revisit the famous Longest Processing Time (*LPT*) rule proposed by Graham in 1969. *LPT* requires to sort jobs in non-ascending order of processing times and then to assign one job at a time to the machine whose load is smallest so far. We provide new insights into LPT and discuss the approximation ratio of a modification of *LPT* that improves Graham’s bound from \(\left( \frac{4}{3} - \frac{1}{3m} \right) \) to \(\left( \frac{4}{3} - \frac{1}{3(m-1)} \right) \) for \(m \ge 3\) and from \(\frac{7}{6}\) to \(\frac{9}{8}\) 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 (\(1,\dots ,m; m+1,\dots ,2m;\) 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.

## Keywords

Identical parallel machine scheduling LPT rule Linear programming Approximation algorithms## Notes

### Acknowledgements

The authors wish to thank an anonymous referee for pointing out the works on the LPT rule by Dosa (2004) and Dosa and Vizvari (2006). This work has been partially supported by “Ministero dell’Istruzione, dell’Università e della Ricerca” Award “TESUN-83486178370409 finanziamento dipartimenti di eccellenza CAP. 1694 TIT. 232 ART. 6.”

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