### Abstract

In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph G such that in the resulting graph G' all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want G' to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial-time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is NP-hard (assuming P ≠ NP). Then, we show that the second problem is APX-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is NP-hard. Finally, we show that the third problem does not admit polynomial-time approximation within a factor of |V |^{1/14?∈} for any ∈ > 0, assuming P ≠ NP (or within a factor of |V

Original language | English |
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Title of host publication | LATIN 2014 |

Subtitle of host publication | Theoretical Informatics - 11th Latin American Symposium, Proceedings |

Publisher | Springer Verlag |

Pages | 683-694 |

Number of pages | 12 |

ISBN (Print) | 9783642544224 |

DOIs | |

Publication status | Published - Jan 1 2014 |

Event | 11th Latin American Theoretical Informatics Symposium, LATIN 2014 - Montevideo, Uruguay Duration: Mar 31 2014 → Apr 4 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 | 8392 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 11th Latin American Theoretical Informatics Symposium, LATIN 2014 |
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Country | Uruguay |

City | Montevideo |

Period | 3/31/14 → 4/4/14 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*LATIN 2014: Theoretical Informatics - 11th Latin American Symposium, Proceedings*(pp. 683-694). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8392 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-642-54423-1_59