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

In their paper titled "Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs," Christine Dahn, Nils M. Kriege, Petra Mutzel, and Julian Schilling propose two fixed-parameter tractable algorithms to solve the weighted Max-Cut problem on embedded 1-planar graphs. A graph is considered 1-planar if it can be drawn in the plane with at most one crossing per edge. The proposed algorithms are parameterized by the crossing number k of the given embedding and recursively reduce a 1-planar graph to at most 3^k planar graphs using edge removal and node contraction techniques. Their main algorithm then solves the Max-Cut problem for these planar graphs using FCE-MaxCut, which was introduced by Liers and Pardella [21]. In case of non-negative edge weights, they suggest a variant that allows solving planar instances with any planar Max-Cut algorithm. The authors demonstrate that a maximum cut in a given 1-planar graph can be derived from solutions for its corresponding planar graphs. Their algorithms compute a maximum cut in an embedded weighted 1-planar graph with n nodes and k edge crossings in time O(3^kn^{3/2} log n). This work is an extension of their conference version (available as arXiv:1803.10983), which is currently under review at TCS. The proposed algorithms have significant implications for real world applications such as network design and optimization problems where finding a maximum cut in a graph is crucial.

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

In their paper titled "Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs," Christine Dahn, Nils M. Kriege, Petra Mutzel, and Julian Schilling propose two fixed-parameter tractable algorithms to solve the weighted Max-Cut problem on embedded 1-planar graphs. This work is an extension of their conference version (available as arXiv:1803.10983), which is currently under review at TCS. The proposed algorithms have significant implications for real world applications such as network design and optimization problems where finding a maximum cut in a graph is crucial.

What are 1 Planar Graphs?

A graph is considered 1-planar if it can be drawn in the plane with at most one crossing per edge. In other words, it can be drawn without any crossings or with only one crossing per edge. This type of graph has been studied extensively due to its importance in many real world applications such as VLSI layout design and network visualization.

The Proposed Algorithms

The proposed algorithms are parameterized by the crossing number k of the given embedding and recursively reduce a 1-planar graph to at most 3^k planar graphs using edge removal and node contraction techniques. Their main algorithm then solves the Max-Cut problem for these planar graphs using FCE-MaxCut, which was introduced by Liers and Pardella [21]. In case of nonnegative edge weights, they suggest a variant that allows solving planar instances with any planar Max Cut algorithm. The authors demonstrate that a maximum cut in a given 1–planar graph can be derived from solutions for its corresponding planar graphs. Their algorithms compute a maximum cut in an embedded weighted 1–planer graph with n nodes and k edge crossings in time O(3^kn^{3/2} log n).

Implications

The proposed algorithms have significant implications for real world applications such as network design and optimization problems where finding a maximum cut in a graph is crucial. By providing efficient fixed parameter tractable algorithms to solve this problem on embedded 1–planer graphs, this research opens up new possibilities for tackling complex optimization tasks more efficiently than before possible while still maintaining accuracy of results obtained from traditional methods like branch & bound or dynamic programming approaches used previously to solve these types of problems