Scheduling meets n-fold integer programming

Abstract

Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In this paper, we continue this study and show that several additional cases of fundamental scheduling problems are fixed-parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where $p_{\max }$ is the maximum processing time of a job and $w_{\max }$ is the maximum weight of a job:

• Makespan minimization on uniformly related machines ($Q||C_{\max }$) parameterized by $p_{\max }$,

• Makespan minimization on unrelated machines ($R||C_{\max }$) parameterized by $p_{\max }$ and the number of kinds of machines (defined later),

• Sum of weighted completion times minimization on unrelated machines ($R||\sum w_jC_j$) parameterized by $p_{\max }+w_{\max }$ and the number of kinds of machines,

• The same problem, $R||\sum w_jC_j$, parameterized by the number of distinct job times and the number of machines.