### 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 (Formula Presented.) all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want (Formula Presented.) 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 (Formula Presented.) for any (Formula Presented.), assuming (Formula Presented.) (or within a factor of (Formula Presented.), assuming (Formula Presented.)).

Original language | English |
---|---|

Pages (from-to) | 371-388 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 73 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 15 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- Colorful components
- Exact polynomial-time algorithms
- Graph coloring
- Hardness of approximation

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica*,

*73*(2), 371-388. https://doi.org/10.1007/s00453-014-9926-0

**Algorithmic and Hardness Results for the Colorful Components Problems.** / Adamaszek, Anna; Popa, Alexandru.

Research output: Contribution to journal › Article

*Algorithmica*, vol. 73, no. 2, pp. 371-388. https://doi.org/10.1007/s00453-014-9926-0

}

TY - JOUR

T1 - Algorithmic and Hardness Results for the Colorful Components Problems

AU - Adamaszek, Anna

AU - Popa, Alexandru

PY - 2014/8/15

Y1 - 2014/8/15

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 (Formula Presented.) all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want (Formula Presented.) 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 (Formula Presented.) for any (Formula Presented.), assuming (Formula Presented.) (or within a factor of (Formula Presented.), assuming (Formula Presented.)).

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 (Formula Presented.) all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want (Formula Presented.) 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 (Formula Presented.) for any (Formula Presented.), assuming (Formula Presented.) (or within a factor of (Formula Presented.), assuming (Formula Presented.)).

KW - Colorful components

KW - Exact polynomial-time algorithms

KW - Graph coloring

KW - Hardness of approximation

UR - http://www.scopus.com/inward/record.url?scp=84940956834&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940956834&partnerID=8YFLogxK

U2 - 10.1007/s00453-014-9926-0

DO - 10.1007/s00453-014-9926-0

M3 - Article

AN - SCOPUS:84940956834

VL - 73

SP - 371

EP - 388

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 2

ER -