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

Compact formulations and valid inequalities for parallel machine scheduling with conflicts