Abstract
The Max-Cut problem is known to be NP-hard on general graphs, while it can be solved in polynomial time on planar graphs. In this paper, we present a fixed-parameter tractable algorithm for the problem on “almost” planar graphs: Given an n-vertex graph and its drawing with k crossings, our algorithm runs in time \(O(2^k(n+k)^{3/2} \log (n + k))\). Previously, Dahn, Kriege and Mutzel (IWOCA 2018) obtained an algorithm that, given an n-vertex graph and its 1-planar drawing with k crossings, runs in time \(O(3^k n^{3/2} \log n)\). Our result simultaneously improves the running time and removes the 1-planarity restriction.
This work is partially supported by JSPS KAKENHI Grant Numbers JP16H02782, JP16K00017, JP16K16010, JP17H01788, JP18H04090, JP18H05291, JP18K11164, and JST CREST JPMJCR1401.
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Acknowledgements
The authors deeply thank anonymous referees for giving us valuable comments. In particular, one of the referees pointed out a flaw in an early version of Lemma 1, which has been fixed in the current paper.
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Kobayashi, Y., Kobayashi, Y., Miyazaki, S., Tamaki, S. (2019). An Improved Fixed-Parameter Algorithm for Max-Cut Parameterized by Crossing Number. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_27
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