## Abstract

A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)^{k}m∊^{−}^{2})-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ∊. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)^{k}+O^{(log3 k)}m log n whenever ∊^{−}^{1} = k^{O}^{(1)}. Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4^{k}m∊^{−}^{2})-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results. We present a deterministic 4^{k}+O(√k^{(log2 k+log2 ∊−1))}m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. Additionally, we present a randomized 4^{k}+O(log k(log k+log ∊^{−}^{1))}m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method. Thus, the algorithm by Brand et al. runs in time 4^{k}+o(k^{)}m whenever ∊^{−}^{1} = 2^{o}(k^{)}, while our deterministic and randomized algorithms run in time 4^{k}+o(k^{)}m log n whenever ∊^{−}^{1} = 2^{o}(k 4 ^{)} and 1 ∊^{−}^{1} = 2^{o}^{(log k k )}, respectively. Prior to our work, no 2^{O}(k^{)}n^{O}^{(1)}-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth.

Original language | English |
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Title of host publication | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |

Editors | Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771092 |

DOIs | |

State | Published - 1 Jul 2019 |

Event | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece Duration: 9 Jul 2019 → 12 Jul 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 132 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |
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Country/Territory | Greece |

City | Patras |

Period | 9/07/19 → 12/07/19 |

## Keywords

- Approximate counting
- K-Path
- Parameterized complexity

## ASJC Scopus subject areas

- Software