## 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 |
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Pages (from-to) | 371-388 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 73 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 5 2015 |

## Keywords

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

## ASJC Scopus subject areas

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