Covering Many (or Few) Edges with k Vertices in Sparse Graphs

TL;DR

This paper investigates fixed-cardinality vertex subset problems in sparse graphs, providing kernelization algorithms and lower bounds, and reveals surprising similarities in their kernelization approaches.

Contribution

It offers a comprehensive analysis of the parameterized complexity of these problems on sparse graphs, including new kernelization algorithms and lower bounds.

Findings

Kernelization algorithms for the problems on sparse graphs.

Kernel lower bounds established for these problems.

Partial Vertex Cover and Max (k,n-k)-Cut share identical kernelization methods.

Abstract

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed $\alpha$ between zero and one we are given a graph and two numbers $k \in \mathbb{N}$ and $t \in \mathbb{Q}$. The task is to find a vertex subset $S$ of exactly $k$ vertices that has value at least (resp. at most for minimization) $t$. Here, the value of a vertex set computes as $\alpha$ times the number of edges with exactly one endpoint in $S$ plus $1-\alpha$ times the number of edges with both endpoints in $S$. These two problems generalize many prominent graph problems, such as Densest $k$-Subgraph, Sparsest $k$-Subgraph, Partial Vertex Cover, and Max ($k$,$n-k$)-Cut. In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max $(k,n-k)$-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.