## Abstract

In this paper, we prove “black box” results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the fine-grained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691).

Read More: https://epubs.siam.org/doi/10.1137/1.9781611975994.135

Read More: https://epubs.siam.org/doi/10.1137/1.9781611975994.135

Original language | English |
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Title of host publication | Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms |

Editors | Shuchi Chawla |

Number of pages | 11 |

Publisher | Society for Industrial and Applied Mathematics |

Publication date | 2020 |

Pages | 2201-2211 |

ISBN (Electronic) | 978-1-61197-599-4 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- approximate counting
- sampling algorithms
- k-hypergraphs
- oracle access
- fine-grained complexity