A Simpler Primal-Dual Proof of Lawler’s Algorithm

The problem of minimizing the weighted sum of completion times when scheduling jobs on a single machine subject to precedence constraints is NP-hard, but can be solved in polynomial time for interesting families of constraints, such as series-parallel constraints and two-dimensional constraints. A classical decomposition theorem of Sydney leads to a simple polynomial-time algorithm for series-parallel constraints, but this is not particularly efficient since a maximum flow computation is needed to compute the decomposition. In contrast, an efficient O(n logn) algorithm for scheduling with series-parallel constraints was given by Lawler in 1978, whose proof is very intricate and subtle. Goemans and Williamson gave a primal-dual proof of Lawler’s result based on a LP-formulation due to Queyranne and Wang. We give an alternate primal-dual proof which is much simpler and is based on the more natural ChudakHochbaum formulation. After that, we turn our attention to two-dimensional precedence constraints, simplifying the recent result by Ambühl and Matrolilli, by proving a stronger structural characterization of a min-cut based approach to compute the optimal solution. [email protected]. Dept. of Computer Science, Cornell University, Ithaca, NY 14853. [email protected]. School of ORIE and Dept. of Computer Science, Cornell University, Ithaca, NY 14853. Research supported partially by NSF grants CCR-0635121, DMI-0500263, DMS-0732196, and CCR-0832782.