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Parameterized Algorithms for Graph Partitioning Problems

  • Hadas Shachnai
  • Meirav ZehaviEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

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.

Our main result is an \(O^*(4^{k+o(k)}\varDelta ^k)\) algorithm for any problem in this class, where \(\varDelta \ge 1\) 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 \(O^*(4^{p+o(p)})\) time algorithm for Max \((k,n-k)\)-Cut, thus improving significantly the best known \(O^*(p^p)\) time algorithm.

Keywords

Time Algorithm Vertex Cover Input Graph Connected Subgraph Blue Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

We thank the anonymous referees for valuable comments and suggestions.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer Science, TechnionHaifaIsrael

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