, Volume 72, Issue 3, pp 734–757 | Cite as

Max-Cut Parameterized Above the Edwards-Erdős Bound

  • Robert Crowston
  • Mark Jones
  • Matthias Mnich


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 2 O(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.


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



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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceRoyal HollowayEghamUK
  2. 2.Cluster of Excellence MMCISaarbrückenGermany

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