When is Red-Blue Nonblocker Fixed-Parameter Tractable?

Serge Gaspers, Joachim Gudmundsson, Michael Horton, Stefan Rümmele

Abstract: In the Red-Blue Nonblocker problem, the input is a bipartite graph \(G=(R \uplus B, E)\) and an integer k, and the question is whether one can select at least k vertices from R so that every vertex in B has a neighbor in R that was not selected. While the problem is W[1]-complete for parameter k, a related problem, Nonblocker, is FPT for parameter k. In the Nonblocker problem, we are given a graph H and an integer k, and the question is whether one can select at least k vertices so that every selected vertex has a neighbor that was not selected. There is also a simple reduction from Nonblocker to Red-Blue Nonblocker, creating two copies of the vertex set and adding an edge between two vertices in different copies if they correspond to the same vertex or to adjacent vertices. We give FPT algorithms for Red-Blue Nonblocker instances that are the result of this transformation – we call these instances symmetric. This is not achieved by playing back the entire transformation, since this problem is NP-complete, but by a kernelization argument that is inspired by playing back the transformation only for certain well-structured parts of the instance. We also give an FPT algorithm for almost symmetric instances, where we assume the symmetry relation is part of the input.