Revisiting Connected Vertex Cover: FPT Algorithms and Lossy Kernels

Abstract

The Connected Vertex Cover problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al. [STOC 2017], have shown an α-approximate kernel for the problem for every α > 1, in the framework of approximate or lossy kernelization. In this work, we exhibit lossy kernels and FPT algorithms for Connected Vertex Cover for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. Our first result is a lossy kernel for Connected Vertex Cover parameterized by the size of a split deletion set. Let n denote the number of vertices in the input graph. We show that

• Connected Vertex Cover parameterized by the size k of a split deletion set admits an α-approximate kernel with $\mathcal {O}(k+(2k+\lceil \frac {2\alpha -1}{\alpha -1} \rceil )^{\lceil \frac {2\alpha -1}{\alpha -1} \rceil })$ vertices and an algorithm with $\mathcal {O}(3^{k} n^{\mathcal {O}(1)})$ running time.

• For the special case when the split deletion set is a clique deletion set, the algorithm runs in ${\mathcal O}(2^{k} n^{{\mathcal O}(1)})$ time and the lossy kernel has $\mathcal {O}(k + \lceil \frac {2\alpha -1}{\alpha -1} \rceil )$ vertices.

To the best of our knowledge, this (approximate) kernel is one of the few lossy kernels for problems parameterized by a structural parameter (that is not solution size). We extend this lossy kernelization to Connected Vertex Cover parameterized by an incomparable parameter, and that is the size of a clique cover. We show that

• Connected Vertex Cover parameterized by the size k of a clique cover is W[1]-hard but admits an α-approximate kernel with $\mathcal {O}(k \lceil \frac {2\alpha -1}{\alpha -1} \rceil )$ vertices for every α > 1. This is one of the few problems that are not FPT but admit a lossy kernel.

Finally, we consider the size of a cluster deletion set as parameter. We show that

• Connected Vertex Cover parameterized by the size k of a cluster deletion set is FPT via an algorithm with running time ${\mathcal O}(4^{k} n^{{\mathcal O}(1)})$. It also admits an α-approximate kernel with $\mathcal {O}(k^{2}+\lceil \frac {2\alpha -1}{\alpha -1} \rceil \cdot \frac {k}{\alpha -1}+ \lceil \frac {\alpha }{\alpha -1} \rceil \cdot k^{\lceil \frac {\alpha }{\alpha -1} \rceil })$ vertices for every α > 1.

• For the special case when the cluster deletion set is a degree-1 modulator, the FPT algorithm runs in ${\mathcal O}(3^{k} n^{{\mathcal O}(1)})$ time and the lossy kernel has $\mathcal {O}(k^{2}+\frac {k}{\alpha -1}+ \lceil \frac {\alpha }{\alpha -1} \rceil \cdot k^{\lceil \frac {\alpha }{\alpha -1} \rceil })$ vertices.