### 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 | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 683-694 |

Number of pages | 12 |

Volume | 8392 LNCS |

ISBN (Print) | 9783642544224 |

DOIs | |

Publication status | Published - 2014 |

Externally published | Yes |

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) |
---|---|

Volume | 8392 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

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

Country | Uruguay |

City | Montevideo |

Period | 3/31/14 → 4/4/14 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 8392 LNCS, 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

**Algorithmic and hardness results for the colorful components problems.** / Adamaszek, Anna; Popa, Alexandru.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 8392 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8392 LNCS, Springer Verlag, pp. 683-694, 11th Latin American Theoretical Informatics Symposium, LATIN 2014, Montevideo, Uruguay, 3/31/14. https://doi.org/10.1007/978-3-642-54423-1_59

}

TY - GEN

T1 - Algorithmic and hardness results for the colorful components problems

AU - Adamaszek, Anna

AU - Popa, Alexandru

PY - 2014

Y1 - 2014

N2 - 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

AB - 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

UR - http://www.scopus.com/inward/record.url?scp=84899969783&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899969783&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-54423-1_59

DO - 10.1007/978-3-642-54423-1_59

M3 - Conference contribution

AN - SCOPUS:84899969783

SN - 9783642544224

VL - 8392 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 683

EP - 694

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -