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On Kernels for Covering and Packing ILPs with Small Coefficients

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Parameterized and Exact Computation (IPEC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8894))

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Abstract

This work continues the study of preprocessing for integer linear programs (ILPs) via the notion of kernelization from parameterized complexity. Previous work, amongst others, studied covering and packing ILPs under different parameterizations and restrictions on the sparseness of the constraint matrix. Several fairly restricted cases, e.g., if every variable appears only in a bounded number of constraints, were shown not to admit polynomial kernels, and in fact are even \(\mathsf {W[1]}\)-hard, due to generalizing the Small Subset Sum(\(k\)) problem; this hardness relies on using coefficients of value exponential in the input size. We study these cases more carefully by taking into account also the coefficient size of the ILPs and obtain a finer classification into cases with polynomial kernels and cases that are fixed-parameter tractable (but without polynomial kernel) in addition to the previously established \(\mathsf {W[1]}\)-hardness for unrestricted coefficients.

Supported by the Emmy Noether-program of the German Research Foundation (DFG), KR 4286/1.

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Notes

  1. 1.

    Intuitively, when domains of variables and coefficients are bounded then Covering ILP and Packing ILP often behave similar to related hypergraph and set problems.

  2. 2.

    The paper [7] mentions only polynomial kernelizations but applies to polynomial compressions as well (cf. [3]).

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Authors and Affiliations

  1. Technical University, Berlin, Germany

    Stefan Kratsch & Vuong Anh Quyen

Authors
  1. Stefan Kratsch
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  2. Vuong Anh Quyen
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Correspondence to Vuong Anh Quyen .

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Editors and Affiliations

  1. Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Marek Cygan

  2. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Kratsch, S., Quyen, V.A. (2014). On Kernels for Covering and Packing ILPs with Small Coefficients. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_26

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  • DOI: https://doi.org/10.1007/978-3-319-13524-3_26

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  • Online ISBN: 978-3-319-13524-3

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