Max-Cut Parameterized Above the Edwards-Erdős Bound

Abstract

We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut parameterized above the Edwards-Erdős bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size

$$ \frac{m}{2} + \frac{n-1}{4} + k $$

in time 2O(k)n 4, or decides that no such cut exists.

This answers a long-standing open question from parameterized complexity that has been posed a number of times over the past 15 years.

Our algorithm has asymptotically optimal running time, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size.

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Notes

  1. 1.

    E.g. the definition in by Bandelt and Mulder [1] agrees with our definition of clique-forest, but the definition in Chap. 3 of Diestel’s book [10] has that every block graph is a tree.

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Acknowledgements

We thank Tobias Friedrich and Gregory Gutin for help with the presentation of the results. We thank the anonymous reviewers for many useful comments and suggestions. Part of this research has been supported by an International Joint Grant from the Royal Society.

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Correspondence to Matthias Mnich.

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Crowston, R., Jones, M. & Mnich, M. Max-Cut Parameterized Above the Edwards-Erdős Bound. Algorithmica 72, 734–757 (2015). https://doi.org/10.1007/s00453-014-9870-z

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Keywords

  • Algorithms and data structures
  • Maximum cuts
  • Combinatorial bounds
  • Fixed-parameter tractability