Parameterized algorithms for graph partitioning problems

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 ⊆ V of size k, such that α1m1 + α2m2 is at most (or at least) p, where α1, α2 ∈ R are constants defining the problem, and m1,m2 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. Our main result is an O∗ (4k+o(k)Δk) algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give an O∗(4p+o(p)) time algorithm for Max (k, n−k)-Cut, thus improving significantly the best known O∗(pp) time algorithm.