## Abstract

We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping φ : V_{G} → V_{H} that maximizes the number of edges of G that are mapped to edges of H. This problem encodes various fundamental NP-hardproblems including Maxcut and Max-k-cut. We also consider the multiway uncut problem. We are given a graph G and a set of terminals T ⊆ V_{G}. We want to partition V _{G} into |T| parts, each containing exactly one terminal, so as to maximize the number of edges in EG having both endpoints in the same part. Multiway uncut can be viewed as a special case of prelabeled MGH where one is also given a prelabeling φ′ : U → V_{H}, U ⊆ V _{G}, and the output has to be an extension of φ′. Both MGH and multiway uncut have a trivial 0.5-approximation algorithm. We present a 0.8535-approximation algorithm for multiway uncut based on a natural linear programming relaxation. This relaxation has an integrality gap of 6/7 ≃ 0.8571, showing that our guarantee is almost tight. For maximum graph homomorphism, we show that a (1/2 + ε_{0})-approximation algorithm, for any constant ε_{0}> 0, implies an algorithm for distinguishing between certain average-case instances of the subgraph isomorphism problem that appear to be hard. Complementing this, we give a (1/2 + Ω(1/|H|log|H|))-approximation algorithm.

Original language | American English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 a |

Publisher | Springer Verlag |

Pages | 176-187 |

Number of pages | 12 |

ISBN (Print) | 3540380442, 9783540380443 |

DOIs | |

State | Published - 2006 |

Externally published | Yes |

Event | 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 - Barcelona, Spain Duration: 28 Aug 2006 → 30 Aug 2006 |

### 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 | 4110 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006 |
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Country/Territory | Spain |

City | Barcelona |

Period | 28/08/06 → 30/08/06 |