When is Red-Blue Nonblocker Fixed-Parameter Tractable?

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.

Next, we augment the parameter by $\ell = \left| B \right| /\left| R \right|$. Somewhat surprisingly, Red-Blue Nonblocker is W[1]-hard for the parameter $k+\ell$, but becomes FPT if no vertex in B has degree 1. The FPT algorithm relies on a structural argument where we show that when |R| is large with respect to k and $\ell$, we can greedily compute a red-blue nonblocker of size at least k. The same results also hold if we augment the parameter by $d_R$ instead of $\ell$, where $d_R$ is the average degree of the vertices in R.