Profile Scheduling of Opposing Forests and Level Orders

The question of existence of a schedule of a given length for n unit length tasks on n identical processors subject to precedence constraints is known to be NP-complete [Ullman, J. Comput. System Sci., 10 (1976), pp. 384–393]. For a fixed value of m we present polynomial algorithms to find an optimal schedule for two families of precedence graphs: level orders and opposing forests. In the case of opposing forest our algorithm is a considerable improvement over the algorithm presented in [Garey et al., SIAM J. Alg. Disc. Meth., 4 (1983), pp. 72–93].