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Consider the problem of sequencing n jobs with general precedence constraints on a single machine to minimize total weighted completion time, which is NP-complete. However some class of problems can be solved in polynomial time, i.e .. the problems whose precedence constraints are trees or series parallel. This paper proposes an O(n4) algorithm for implementing a decomposition procedure due to Sidney. It also shows that if each decomposed sub graph is series parallel, then the problem is solved in 0(n ) running time, even if the whole graph is not series parallel.

ALGORITHM FOR ONE MACHINE JOB SEQUENCING WITH PRECEDENCE CONSTRAINTS