A novel integer programing formulation for scheduling with family setup times on a single machine to minimize maximum lateness

To explain this performance, we analyze the theoretical tightness of this formulation. We show that if the number of jobs in each family is bounded then the gap between a heuristic rounding and the lower bound produced by the linear programing increases at most sub-linearly with the number of jobs. The optimality gaps of prior approximation algorithms grow linearly with the number of jobs. Our work improves on these prior results.