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Scheduling Kernels via Configuration LP

Authors Dušan Knop, Martin Koutecký



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Dušan Knop
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Martin Koutecký
  • Computer Science Institute of Charles University, Prague, Czech Republic

Acknowledgements

Authors wish to thank the Lorentz Center for making it possible to organize the workshop Scheduling Meets Fixed-Parameter Tractability, which output resulted in this paper. The contribution of the Lorentz Center in stimulating suggestions, giving feedback and taking care of all practicalities, helped us to focus on our research and to organize a meeting of high scientific quality. Furthermore, the authors thank Stefan Kratsch for bringing Proposition 5 to their attention.

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Dušan Knop and Martin Koutecký. Scheduling Kernels via Configuration LP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 73:1-73:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.73

Abstract

Makespan minimization (on parallel identical or unrelated machines) is arguably the most natural and studied scheduling problem. A common approach in practical algorithm design is to reduce the size of a given instance by a fast preprocessing step while being able to recover key information even after this reduction. This notion is formally studied as kernelization (or simply, kernel) - a polynomial time procedure which yields an equivalent instance whose size is bounded in terms of some given parameter. It follows from known results that makespan minimization parameterized by the longest job processing time p_max has a kernelization yielding a reduced instance whose size is exponential in p_max. Can this be reduced to polynomial in p_max? We answer this affirmatively not only for makespan minimization, but also for the (more complicated) objective of minimizing the weighted sum of completion times, also in the setting of unrelated machines when the number of machine kinds is a parameter. Our algorithm first solves the Configuration LP and based on its solution constructs a solution of an intermediate problem, called huge N-fold integer programming. This solution is further reduced in size by a series of steps, until its encoding length is polynomial in the parameters. Then, we show that huge N-fold IP is in NP, which implies that there is a polynomial reduction back to our scheduling problem, yielding a kernel. Our technique is highly novel in the context of kernelization, and our structural theorem about the Configuration LP is of independent interest. Moreover, we show a polynomial kernel for huge N-fold IP conditional on whether the so-called separation subproblem can be solved in polynomial time. Considering that integer programming does not admit polynomial kernels except for quite restricted cases, our "conditional kernel" provides new insight.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Integer programming
Keywords
  • Scheduling
  • Kernelization

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