Core Concepts
The authors present three tools to improve the running time of algorithms for solving high-multiplicity scheduling problems on identical machines, such as minimizing makespan (Cmax), minimizing minimum load (Cmin), and minimizing the difference between maximum and minimum load (Cenvy).
Abstract
The authors focus on the parameter d, the number of different job types, and aim to develop algorithms with running times that are efficient with respect to this parameter.
The first tool is a balancing result by Govzmann et al. that allows pre-scheduling many of the jobs, bounding the makespan by 2dpmax, where pmax is the largest processing time. This leads to algorithms with running times of (d log(pmax) + log(2dp2max))2O(d)⟨I⟩O(1) for P||{Cmax, Cmin} and pmax(d log(pmax) + log(2dp2max))2O(d)⟨I⟩O(1) for P||Cenvy.
The second tool involves solving a special relaxation of an integer linear program (ILP) associated with the problem's PQ-representation, and then using a proximity result to reduce the coefficients in the system describing the polytope P. This yields an algorithm with running time independent of the right-hand-side values.
The third tool provides a new upper bound for the number of vertices of the integer hull of a polytope, which is used to improve the running time of the algorithm by Jansen and Klein. This also leads to algorithms with running times of the form (log(pmax))2O(d)⟨I⟩O(1).
The authors also show lower bounds for P||Cmax that somewhat resemble the running time of their algorithms, as well as a matching lower bound for the additive approximation scheme.
Finally, the authors establish a connection between the parameterized complexity of P||{Cmax, Cmin} and Q||{Cmax, Cmin}, showing that they have the same answer with respect to fixed-parameter tractability.