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Parameterized Shifted Combinatorial Optimization

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Book cover Computing and Combinatorics (COCOON 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10392))

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Abstract

Shifted combinatorial optimization is a new nonlinear optimization framework which is a broad extension of standard combinatorial optimization, involving the choice of several feasible solutions at a time. This framework captures well studied and diverse problems ranging from so-called vulnerability problems to sharing and partitioning problems. In particular, every standard combinatorial optimization problem has its shifted counterpart, which is typically much harder. Already with explicitly given input set the shifted problem may be NP-hard. In this article we initiate a study of the parameterized complexity of this framework. First we show that shifting over an explicitly given set with its cardinality as the parameter may be in XP, FPT or P, depending on the objective function. Second, we study the shifted problem over sets definable in MSO logic (which includes, e.g., the well known MSO partitioning problems). Our main results here are that shifted combinatorial optimization over MSO definable sets is in XP with respect to the MSO formula and the treewidth (or more generally clique-width) of the input graph, and is W[1]-hard even under further severe restrictions.

J. Gajarský’s research was partially supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (ERC Consolidator Grant DISTRUCT, grant agreement No 648527).

P. Hliněný, and partially J. Gajarský, were supported by the research centre Institute for Theoretical Computer Science (CE-ITI), project P202/12/G061 of the Czech Science Foundation.

M. Koutecký was partially supported by the project 17-09142S of the Czech Science Foundation.

Shmuel Onn was partially supported by the Dresner Chair at the Technion.

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Notes

  1. 1.

    We remark that while [15, Theorem 4] explicitly speaks about generally weaker \(MSO_1\) logic, it is folklore that in the realm of graphs of bounded treewidth the same can be equivalently stated with \(MSO_2\) logic (as noted also in [15]).

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Author notes

  1. Jakub Gajarský

    Present address: Technical University Berlin, Berlin, Germany

Authors and Affiliations

  1. Masaryk University, Brno, Czech Republic

    Jakub Gajarský & Petr Hliněný

  2. Charles University, Prague, Czech Republic

    Martin Koutecký

  3. Technion - Israel Institute of Technology, Haifa, Israel

    Shmuel Onn

Authors
  1. Jakub Gajarský
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  2. Petr Hliněný
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  3. Martin Koutecký
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  4. Shmuel Onn
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Corresponding author

Correspondence to Petr Hliněný .

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Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Texas A&M University, College Station, Texas, USA

    Jianer Chen

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Gajarský, J., Hliněný, P., Koutecký, M., Onn, S. (2017). Parameterized Shifted Combinatorial Optimization. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_19

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  • DOI: https://doi.org/10.1007/978-3-319-62389-4_19

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