🤖 AI Summary
This paper addresses the NP-hard parallel machine scheduling problem with conflict constraints (P||Cₘₐₓ), where conflicting jobs cannot be assigned to the same machine and the objective is to minimize the makespan. We propose a novel, compact mixed-integer linear programming (MILP) formulation grounded in graph coloring. Through polyhedral analysis, we introduce, for the first time, strong valid inequalities from the stable-set polytope into a branch-and-cut framework. Computational evaluation on the most challenging benchmark instances demonstrates that our approach significantly outperforms existing algorithms: it achieves faster solution times and stronger optimality guarantees—particularly when the gap between the trivial lower bound and the optimal makespan widens. The integration of theoretically grounded cutting planes enhances both the tightness of the relaxation and the efficiency of the exact solver.
📝 Abstract
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.
