Structural Results for High-Multiplicity Scheduling on Uniform Machines

Parameterizing by the largest processing time $p_{max}$ and the number of different job processing times $d$, we propose a proximity technique for High-Multiplicity Scheduling on Uniform Machines for the objectives Makespan Minimization ($C_{max}$) and Santa Claus ($C_{min}$) to obtain new structural results for these problems. The novelty in our approach is that we deal with a fractional solution for only a sub-instance, where the sub-instance itself is not known a priori. While the construction and computation of the fractional solution -- in contrast to usual proximity techniques -- is not done in polynomial time, this also allows us to formulate a comparably strong and general proximity statement. Eventually, this allows us to reduce the number of jobs that need to be distributed to a polynomial in $p_{max}$ for each machine and job type, by preassigning jobs according to the fractional solution, essentially returning a bounded number (at most $O(p_{max}^{O(d^2)})$) of kernels, one for each (guessed) sub-instance. We can use our structural results to obtain an algorithm with running time is $p_{max}^{O(d^2)}poly|I|$, matching the best-known so far by Knop et al. (Oper. Res. Lett. '21). Moreover, we propose an $p_{max}^{O(d^2)} poly |I|$ time algorithm for Envy Minimization $C_{envy}$ in the High-Multiplicity Setting on Uniform Machines, showing that this problem is \textsc{fpt} in $p_{max}$. Eventually, we also propose a general mechanism to bound the largest coefficient in the Configuration ILP for so called \emph{Load Balancing Problems} by $(dp_{max})^{O(d)}$, which we hope to be of interest for the development of algorithms.