The problem of scheduling conflicting jobs on parallel machines consists in assigning a set of jobs to a set of machines so that no two conflicting jobs are allocated to the same machine, and the maximum processing time among all machines is minimized. We propose a new compact mixed integer linear formulation based on the representatives model for the vertex coloring problem, which overcomes a number of issues inherent in the natural assignment model. We present a polyhedral study of the associated polytope, and describe classes of valid inequalities inherited from the stable set polytope. We describe branch-and-cut algorithms for the problem, and report on computational experiments with benchmark instances. Our computational results on the hardest instances of the benchmark set show that the proposed algorithms are superior (either in running time or quality of the solutions) to the current state-of-the-art methods. We find that our new method performs better than the existing ones especially when the gap between the optimal value and the trivial lower bound (i.e., the sum of all processing times divided by the number of machines) increases.

翻译：并行机冲突调度问题旨在将一组作业分配到一组机器上，使得任意两个冲突作业不被分配到同一台机器，并最小化所有机器中的最大处理时间。针对该问题，我们基于顶点着色问题的代表元模型，提出了一种新的紧凑混合整数线性规划模型，该模型克服了自然分配模型中固有的若干问题。我们对相关多面体进行了多面体分析，并描述了从稳定集多面体继承的若干类有效不等式。针对该问题，我们设计了分支切割算法，并对基准测试实例进行了计算实验。在基准集中最困难实例上的计算结果表明，所提算法在运行时间或解的质量上均优于当前最先进的方法。我们发现，当最优值与平凡下界（即所有处理时间之和除以机器数量）之间的差距增大时，新方法的性能提升尤为显著。