Parameterized Complexity of Feedback Vertex Set with Connectivity Constraints

Abstract

The Feedback Vertex Set (FVS) problem, together with several of its variants, is arguably one of the most well-studied problems in the field of Parameterized Complexity. Two versions of the problem that have garnered significant interest involve the inclusion of an independence constraint and a connectivity constraint in the solution. This paper introduces generalized versions of both these variants, known as At least-c-FVS and At most-c-FVS, respectively, serving as extensions of Connected FVS and Independent FVS, respectively. The problem At most-c-FVS (resp., At least-c-FVS) is defined as follows: given a graph G and an integer k, the objective is to determine whether there exists a subset $S \subseteq V(G)$, with $|S| \le k$, such that the subgraph $G-S$ is a forest and each component of G[S] contains at most c (resp., at least c) vertices. We study these problems in the realm of Parameterized Complexity and obtain the following results:

– At most-c-FVS parameterized by k has a kernel of size $\mathcal {O}(k^{3+c})$, and admits an FPT algorithm running in time $2^{\mathcal {O}(k) +c \cdot \log (k^2)} \cdot n^{\mathcal {O}(1)}$.

– At least-c-FVS parameterized by k has no kernel of size $k^{f(c)}$ for any computable function f (unless co-NP $\subseteq$ NP/poly), but admits an FPT algorithm running in time $2^{\mathcal {O}(k)}\cdot n^{\mathcal {O}(1)}$.

S. Jana—Supported by the Engineering and Physical Sciences Research Council (EPSRC) via the project MULTIPROCESS (grant no. EP/V044621/1)

S. Saurabh—Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416); and he also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.