Covering Many (or Few) Edges with k Vertices in Sparse Graphs
Tomohiro Koana, Christian Komusiewicz, Andr\'e Nichterlein, Frank, Sommer

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 between zero and one we are given a graph and two numbers and . The task is to find a vertex subset of exactly vertices that has value at least (resp. at most for minimization) . Here, the value of a vertex set computes as times the number of edges with exactly one endpoint in plus times the number of edges with both endpoints in . These two problems generalize many prominent graph problems, such as Densest -Subgraph, Sparsest -Subgraph, Partial Vertex Cover, and Max (,)-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…
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