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The Minimum Feasible Tileset Problem

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Abstract

We introduce and study the Minimum Feasible Tileset problem: given a set of symbols and subsets of these symbols (scenarios), find a smallest possible number of pairs of symbols (tiles) such that each scenario can be formed by selecting at most one symbol from each tile. We show that this problem is \(\mathsf {APX}\)-hard and that it is \(\mathsf {NP}\)-hard even if each scenario contains at most three symbols. Our main result is a 4/3-approximation algorithm for the general case. In addition, we show that the Minimum Feasible Tileset problem is fixed-parameter tractable both when parameterized with the number of scenarios and with the number of symbols.

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Notes

  1. Recall that \(\mathcal {O}^*\)-notation ignores factors that are polynomial in the input size.

  2. A compression to \(\mathcal {O}(|F|^d)\) size can be achieved by specifying one bit for each possible scenario in \(\mathcal {S} \) and setting it to one if the scenario is present and zero otherwise.

  3. Dell and Marx called this problem Perfect d-Set Matching.

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Author information

Authors and Affiliations

  1. Institut für Mathematik, Graduate School CE, TU Darmstadt, Darmstadt, Germany

    Yann Disser

  2. Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany

    Stefan Kratsch

  3. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    Manuel Sorge

  4. Department of Industrial Engineering and Management, Ben Gurion University of the Negev, Beer Sheva, Israel

    Manuel Sorge

Authors
  1. Yann Disser
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  2. Stefan Kratsch
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  3. Manuel Sorge
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Corresponding author

Correspondence to Manuel Sorge.

Additional information

An extended abstract of this article appeared at the 12th Workshop on Approximation and Online Algorithms, Wrocław, September 2014 [10]. In comparison, apart from full proof details, the present article additionally contains examples and an APX-hardness proof. The authors gratefully acknowledge support by the Alexander von Humboldt-Foundation (Yann Disser), the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School CE at TU Darmstadt (Yann Disser), the German Research Foundation (DFG), Projects KR 4286/1 (Stefan Kratsch) and NI 369/12 (Manuel Sorge), the Israel Science Foundation, Grant No. 551145/14 (Manuel Sorge), and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement Number 631163.11 (Manuel Sorge).

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Disser, Y., Kratsch, S. & Sorge, M. The Minimum Feasible Tileset Problem. Algorithmica 81, 1126–1151 (2019). https://doi.org/10.1007/s00453-018-0460-3

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  • DOI: https://doi.org/10.1007/s00453-018-0460-3

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