Parameterized Algorithms for Graph Partitioning Problems

Hadas Shachnai, Meirav Zehavi

Abstract: We study a broad class of graph partitioning problems, where each problem is specified by a graph \(G=(V,E)\), and parameters \(k\) and \(p\). We seek a subset \(U\subseteq V\) of size \(k\), such that \(\alpha _1m_1 + \alpha _2m_2\) is at most (or at least) \(p\), where \(\alpha _1,\alpha _2\in \mathbb {R}\) are constants defining the problem, and \(m_1, m_2\) are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in \(U\), respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max \((k,n-k)\)-Cut, Min \(k\)-Vertex Cover, \(k\)-Densest Subgraph, and \(k\)-Sparsest Subgraph.