### Abstract

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^{k}, for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. This completely settles the question regarding the complexity of modular computation of the number of spanning trees in planar graphs. The techniques used rely heavily on algebraic-topology. In the spirit of counting problems modulo 2^{k}, we also exhibit a highly parallel ⊕L algorithm for finding the value of a Permanent modulo 2^{k}. Previously, the best known result in this direction was Valiant's result that this problem lies in P.

Original language | English (US) |
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Title of host publication | Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007 |

Pages | 222-235 |

Number of pages | 14 |

DOIs | |

State | Published - Oct 2 2007 |

Externally published | Yes |

Event | 22nd Annual IEEE Conference on Computational Complexity, CCC 2007 - San Diego, CA, United States Duration: Jun 13 2007 → Jun 16 2007 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
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ISSN (Print) | 1093-0159 |

### Other

Other | 22nd Annual IEEE Conference on Computational Complexity, CCC 2007 |
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Country | United States |

City | San Diego, CA |

Period | 6/13/07 → 6/16/07 |

### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science
- Computational Mathematics

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## Cite this

*Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007*(pp. 222-235). [4262766] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2007.23