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.
|Title of host publication||Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers|
|Editors||Dieter Kratsch, Ioan Todinca|
|Number of pages||12|
|State||Published - 1 Jan 2014|
|Event||40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014 - Orléans, France|
Duration: 25 Jun 2014 → 27 Jun 2014
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014|
|Period||25/06/14 → 27/06/14|