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Saving Colors and Max Coloring: Some Fixed-Parameter Tractability Results

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Graph-Theoretic Concepts in Computer Science (WG 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9941))

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Abstract

Max coloring is a well known generalization of the usual Min Coloring problem, widely studied from (standard) complexity and approximation viewpoints. Here, we tackle this problem under the framework of parameterized complexity. In particular, we first show to what extend the result of [3] - saving colors from the trivial bound of n on the chromatic number - extends to Max Coloring. Then we consider possible improvements of these results by considering the problem of saving colors/weight with respect to a better bound on the chromatic number. Finally, we consider the fixed parameterized tractability of Max Coloring in restricted graph classes under standard parameterization.

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Notes

  1. 1.

    In batch scheduling, a color is a set of tasks (batch) processed together; then the weight of the color is the time to process the batch, and the total weight is the time to process every batch, i.e., every task.

  2. 2.

    A classical reduction from 3-SAT produces a graph with \(\varTheta (var+clauses)\) vertices and edges, where var and clauses are the number of variables and clauses, see for instance http://cgi.csc.liv.ac.uk/~igor/COMP309/3CP.pdf.

  3. 3.

    We use \(v'_i\) to denote vertices of \(G'\) (so \(v'_i=v_i\) for \(i<\ell \) and \(v'_i=v_{i+1}\) for \(i\ge \ell \)).

  4. 4.

    Thanks to Step 2, each color of size 1 is adjacent to all other colors, so we can reorder colors if needed to put colors of size 1 at the end while preserving the fact that each vertex in \(S_j\) is adjacent to each color \(S_i\), \(i<j\).

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6 UMR 7606, 4 Place Jussieu, 75005, Paris, France

    Bruno Escoffier

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  1. Bruno Escoffier
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Correspondence to Bruno Escoffier .

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Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Escoffier, B. (2016). Saving Colors and Max Coloring: Some Fixed-Parameter Tractability Results. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_5

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  • DOI: https://doi.org/10.1007/978-3-662-53536-3_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53535-6

  • Online ISBN: 978-3-662-53536-3

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