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The best method known for determining lower bounds on the vertex coloring number of a graph is the linear-programming column-generation technique, where variables correspond to stable sets, first employed by Mehrotra and Trick in 1996. We present an implementation of the method that provides numerically-safe results, independent of the floating-point accuracy of linear-programming software. Our work includes an improved branch-andbound algorithm for maximum-weight stable sets and a parallel branch-andprice framework for graph coloring. Computational results are presented on a collection of standard test instances, including the unsolved challenge problems created by David S. Johnson in 1989.

In this paper we compare the static and dynamic application of heuristic and optimal solution methods to job‐shop scheduling problems when processing times are uncertain. Recently developed optimizing algorithms and several heuristics are used to evaluate 53 standard job‐shop scheduling problems with a makespan objective when job processing times are known with varying degrees of uncertainty. Results indicate that fixed optimal sequences derived from deterministic assumptions quickly deteriorate with the introduction of processing time uncertainty when compared with dynamically updated heuristic schedules. As processing time uncertainty grows, we demonstrate that simple dispatch heuristics provide performance comparable or superior to that of algorithmically more sophisticated scheduling policies.

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