## Abstract

We present a new approximation algorithm for the treewidth problem which constructs a corresponding tree decomposition as well. Our algorithm is a faster variation of Reed’s classical algorithm. For the benefit of the reader, and to be able to compare these two algorithms, we start with a detailed time analysis for Reed’s algorithm. We fill in many details that have been omitted in Reed’s paper. Computing tree decompositions parameterized by the treewidth k is fixed parameter tractable (FPT), meaning that there are algorithms running in time O(f(k)g(n)) where f is a computable function, g(n) is polynomial in n, and n is the number of vertices. An analysis of Reed’s algorithm shows f(k) = 2^{O} ^{(} ^{k} ^{log} ^{k} ^{)} and g(n) = nlog n for a 5-approximation. Reed simply claims time O(nlog n) for bounded k for his constant factor approximation algorithm, but the bound of 2^{Ω} ^{(} ^{k} ^{log} ^{k} ^{)}nlog n is well known. From a practical point of view, we notice that the time of Reed’s algorithm also contains a term of O(k^{2}2^{24} ^{k}nlog n), which for small k is much worse than the asymptotically leading term of 2^{O} ^{(} ^{k} ^{log} ^{k} ^{)}nlog n. We analyze f(k) more precisely, because the purpose of this paper is to improve the running times for all reasonably small values of k. Our algorithm runs in O(f(k) nlog n) too, but with a much smaller dependence on k. In our case, f(k) = 2^{O} ^{(} ^{k} ^{)}. This algorithm is simple and fast, especially for small values of k. We should mention that Bodlaender et al. [2016] have an asymptotically faster algorithm running in time 2^{O} ^{(} ^{k} ^{)}n. It relies on a very sophisticated data structure and does not claim to be useful for small values of k.

Original language | English (US) |
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Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - 15th International Conference and Workshops, WALCOM 2021, Proceedings |

Editors | Ryuhei Uehara, Seok-Hee Hong, Subhas C. Nandy |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 166-181 |

Number of pages | 16 |

ISBN (Print) | 9783030682101 |

DOIs | |

State | Published - 2021 |

Event | 15th International Conference on Algorithms and Computation, WALCOM 2021 - Virtual, Online Duration: Feb 28 2021 → Mar 2 2021 |

### 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 | 12635 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 15th International Conference on Algorithms and Computation, WALCOM 2021 |
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City | Virtual, Online |

Period | 2/28/21 → 3/2/21 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science