Deconstructing Parameterized Hardness of Fair Vertex Deletion Problems

For a graph G and a positive integer d, a set S ⊆ V (G) is a fair set with the fairness factor d if for every vertex in G, at most d of its neighbors are in the S. In the Π -Vertex Deletion problem, the aim is to find in a given graph a set S of minimum size such that G\S satisfies the property Π. We look at Π -Fair Vertex Deletion problem where we also want S to be a fair set. It is known that the general Π -Fair Vertex Deletion problem where Π is expressible by a first order formula and is also given as input is W[1]-hard when parameterized by treewidth. Our first observation is that if we also parameterize by the fairness constant d, then Π -Fair Vertex Deletion is FPT (fixed-parameter tractable) even if Π -Vertex Deletion can be expressed by a Monadic Second Order (MSO) formula. As a corollary we get an FPT algorithm for Fair Feedback Vertex Set (FFVS) and Fair Vertex Cover (FVC) parameterized by solution size. We then do a deep dive on FVC and more generally Π -Fair Vertex Deletion problems parameterized by solution size k when Π is characterized by a finite set of forbidden graphs. We show that these problems are FPT and develop a polynomial kernel when d is a constant. While the FPT algorithms use the standard branching technique, the fairness constraint introduces challenges to design a polynomial kernel. En route, we give a polynomial kernel for a special instance of Min-ones-SAT and Fair q-hitting set, a generalization of q-hitting set, with a fairness constraint on an underlying graph structure on the universe. These could be of independent interest. To complement our FPT results, we show that Fair Set and Fair Independent Set problems are W[1]-hard even in 3-degenerate graphs when the fairness factor is 1. We also show that FVC is polynomial-time solvable when d = 1 or 2, and NP -hard for d ≥ 3, and that FFVS is NP -hard for all d ≥ 1.