### Abstract

For a fixed digraph H, the minimum cost homomorphism problem, MinHOM(H), asks whether an input digraph G, with given costs c _{i} (u), u∈ ∈V(G), i∈ ∈V(H), and an integer k, admits a homomorphism to H of total cost not exceeding k. Minimum cost homomorphism problems encompass many well studied optimization problems such as list homomorphism problems, retraction and precolouring extension problems, chromatic partition optimization, and applied problems in repair analysis. For undirected graphs the complexity of the problem, as a function of the parameter H, is well understood; for digraphs, the situation appears to be more complex, and only partial results are known. We focus on the minimum cost homomorphism problem for reflexive digraphs H. It is known that MinHOM(H) is polynomial if H has a Min-Max ordering. We prove that for any other reflexive digraph H, the problem MinHOM(H) is NP-complete. (This was earlier conjectured by Gutin and Kim.) Apart from undirected graphs, this is the first general class of digraphs for which such a dichotomy has been proved. Our proof involves a forbidden induced subgraph characterization of reflexive digraphs with a Min-Max ordering, and implies a polynomial test for the existence of a Min-Max ordering in a reflexive digraph H.

Original language | English |
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Title of host publication | LATIN 2008: Theoretical Informatics - 8th Latin American Symposium, Proceedings |

Pages | 182-193 |

Number of pages | 12 |

Volume | 4957 LNCS |

DOIs | |

Publication status | Published - 2008 |

Externally published | Yes |

Event | 8th Latin American TheoreticalINformatics Symposium, LATIN 2008 - Buzios, Brazil Duration: 7 Apr 2008 → 11 Apr 2008 |

### 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 | 4957 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 8th Latin American TheoreticalINformatics Symposium, LATIN 2008 |
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Country | Brazil |

City | Buzios |

Period | 7/4/08 → 11/4/08 |

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### Keywords

- Dichotomy
- Homomorphism
- Minimum cost homomorphism
- NP-completeness
- Polynomial time algorithm
- Reflexive digraph

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*LATIN 2008: Theoretical Informatics - 8th Latin American Symposium, Proceedings*(Vol. 4957 LNCS, pp. 182-193). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4957 LNCS). https://doi.org/10.1007/978-3-540-78773-0_16