## 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 α_{1}m_{1} + α_{2}m_{2} is at most (or at least) p, where α_{1}, α_{2} ∈ 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)}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∗(4^{p+o}(p)) time algorithm for Max (k, n−k)-Cut, thus improving significantly the best known O∗(p^{p}) time algorithm.

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers |

Editors | Dieter Kratsch, Ioan Todinca |

Publisher | Springer Verlag |

Pages | 384-395 |

Number of pages | 12 |

ISBN (Electronic) | 9783319123394 |

DOIs | |

State | Published - 1 Jan 2014 |

Externally published | Yes |

Event | 40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014 - Orléans, France Duration: 25 Jun 2014 → 27 Jun 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8747 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014 |
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Country/Territory | France |

City | Orléans |

Period | 25/06/14 → 27/06/14 |