### Abstract

Vassilevska and Williams (STOC'09) showed how to count simple paths on k vertices and matchings on k/2 edges in ann-vertex graph in time nk/2+O(1). In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP'09), and Bjorklund et al. (ESA'09), via nst/2+O(1)-time algorithms for counting t-tuples of pairwise disjoint sets drawn from a given family of s-sized subsets of an n-element universe. Shortly afterwards, Alon and Gutner (TALG'10) showed that these problems have O(n st/2) and O(nk/2) lower bounds when counting by color coding. Here, we show that one can do better-we show that the "meet-in-the-middle" exponent st/2 can be beaten and give an algorithm that counts in time n0.45470382st+O(1) for t a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on k vertices and pathwidth pk in an n-vertex graph in n0.45470382k+2p+O(1) time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound. We also give improved bounds for counting t-tuples of disjoint s-sets for s = 2, 3, 4. Our algorithms use fast matrix multiplication. We show an argument that this is necessary to go below the meet-in-the-middle barrier.

Original language | English |
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Article number | 48 |

Pages (from-to) | 1-26 |

Journal | ACM Transactions on Algorithms |

Volume | 13 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Sep 2017 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- Counting low pathwidth graphs
- Counting matchings
- Counting packings
- Counting paths
- FPT algorithms
- Matrix multiplication
- Meet in the middle

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## Cite this

*ACM Transactions on Algorithms*,

*13*(4), 1-26. [48]. https://doi.org/10.1145/3125500