Fair Vertex Problems Parameterized by Cluster Vertex Deletion
Tomáš Masařík, Jędrzej Olkowski, Anna Zych-Pawlewicz
TL;DR
This work analyzes fair vertex problems under structural parameterization, focusing on keeping the solution distributed across neighborhoods via the fair cost $fc(X)$. It proves a hardness barrier: the general fair-VE metatheorem for FO deletion parameterized by cluster vertex deletion and formula size is W[1]-hard, but identifies a powerful positive regime based on coherent shapes that yields an MSO$_1$-FairVE FPT algorithm, using shapes and a Lenstra-type ILP to compute minimal fair-cost solutions. The framework applies to a broad class of problems, including Fair Vertex Cover, Fair Feedback Vertex Set, Fair Odd Cycle Transversal, Fair Dominating Set, and Fair $[\sigma,\rho]$-Domination under specific settings, unifying them under a common model-checking and ILP-based approach. This advances understanding of which fair-vertex problems remain tractable on dense graphs and suggests directions for extending FPT results to other graph parameters such as modular-width or clique-width. The methods combine logical model-checking with combinatorial deconstructions (modulators, neighborhood types) and integer programming, offering both theoretical insights and practical algorithmic templates.
Abstract
We study fair vertex problem metatheorems on graphs within the scope of structural parameterization in parameterized complexity. Unlike typical graph problems that seek the smallest set of vertices satisfying certain properties, the goal here is to find such a set that does not contain too many vertices in any neighborhood of any vertex. Formally, the task is to find a set $X\subseteq V(G)$ of fair cost $k$, i.e., such that for all $v\in V(G)$ $|X\cap N(v)|\le k$. Recently, Knop, Masařík, and Toufar [MFCS 2019] showed that all fair MSO$_1$ definable problems can be solved in FPT time parameterized by the twin cover of a graph. They asked whether such a statement would be achievable for a more general parameterization of cluster vertex deletion, i.e., the smallest number of vertices required to be removed from the graph such that what remains is a collection of cliques. In this paper, we prove that in full generality this is not possible by demonstrating a W[1]-hardness. On the other hand, we give a sufficient property under which a fair MSO$_1$ definable problem admits an FPT algorithm parameterized by the cluster vertex deletion number. Our algorithmic formulation is very general as it captures the fair variant of many natural vertex problems such as the Fair Feedback Vertex Set, the Fair Vertex Cover, the Fair Dominating Set, the Fair Odd Cycle Transversal, as well as a connected variant of thereof. Moreover, we solve the Fair $[σ,ρ]$-Domination problem for $σ$ finite, or $σ=\mathbb{N}$ and $ρ$ cofinite. Specifically, given finite or cofinite $ρ\subseteq \mathbb{N}$ and finite $σ$, or $ρ\subseteq \mathbb{N}$ cofinite and $σ=\mathbb{N}$, the task is to find set of vertices $X\subseteq V(G)$ of fair cost at most $k$ such that for all $v\in X$, $|N(v)\cap X|\inσ$ and for all $v\in V(G)\setminus X$, $|N(v)\cap X|\inρ$.