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arXiv · 1812.03074

Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs

Christine Dahn·Nils M. Kriege·Petra Mutzel·Julian Schilling
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Abstract

We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with $n$ nodes and $k$ edge crossings in time $O(3^k \cdot n^{3/2} \log n)$.

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Christine Dahn, Nils M. Kriege, Petra Mutzel, Julian Schilling. 2018-11-29. Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs. https://arxiv.org/abs/1812.03074

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