Maximum Cut Parameterized by Crossing Number [article]

Markus Chimani, Christine Dahn, Martina Juhnke-Kubitzke, Nils M. Kriege, Petra Mutzel, Alexander Nover
2019 arXiv   pre-print
Given an edge-weighted graph G on n nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number k of crossings in a given drawing of G. Our algorithm achieves a running time of O(2^k p(n + k)), where p is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs
more » ... , at most one crossing per edge) and its dependency on k is of order 3^k . A direct consequence of our result is that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number.
arXiv:1903.06061v2 fatcat:cc7ynjtixrfehisgsxbmbrhapq