# 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.

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1. 1.

Theorem 4.1 does not explicitly mention the step length $$\gamma$$, but it is implicit in the proof that even this stronger statement holds. Also, Theorem 4.1 does not give the complexity bound, which is contained in Theorem 6.2.)

2. 2.

For example, if, for any $$\alpha \in \mathbb {R}$$, there is a separation oracle for the level set $$\{\mathbf{x}\mid f(\mathbf{x}) \le \alpha \}$$.

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## Acknowledgements

We would like to thank René van Bevern for pointing us to much related work. We are grateful to Matthias Mnich for pointing out the exponential speed-up of our result compared to the previous work, and other useful comments. Finally, we are grateful to the anonymous reviewers for their comments which made the paper more readable.

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Correspondence to Dušan Knop.