### Abstract

Valiant (SIAM Journal on Computing 8, pages 410-421) showed that the problem of counting the number of s-t paths in graphs (both in the case of directed graphs and in the case of undirected graphs) is complete for #P under polynomial-time one-Turing reductions (namely, some post-computation is needed to recover the value of a #P-function). Valiant then asked whether the problem of counting the number of self-avoiding walks of length n in the two-dimensional grid is complete for #P1, i.e., the tally-version of #P. This paper offers a partial answer to the question. It is shown that a number of versions of the problem of computing the number of self-avoiding walks in twodimensional grid graphs (graphs embedded in the two-dimensional grid) is polynomial-time one-Turing complete for #P. This paper also studies the problem of counting the number of selfavoiding walks in graphs embedded in a hypercube. It is shown that a number of versions of the problem is polynomial-time one-Turing complete for #P, where a hypercube graph is specified by its dimension, a list of its nodes, and a list of its edges. By scaling up the completeness result for #P, it is shown that the same variety of problems is polynomial-time one-Turing complete for #EXP, where the post-computation required is right bit-shift by exponentially many bits and a hypercube graph is specified by: its dimension, a boolean circuit that accept its nodes, and one that accepts its edges.

Original language | English (US) |
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Title of host publication | Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings |

Publisher | Springer Verlag |

Pages | 585-597 |

Number of pages | 13 |

Volume | 2136 |

ISBN (Print) | 9783540446835 |

State | Published - 2001 |

Externally published | Yes |

Event | 26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001 - Marianske Lazne, Czech Republic Duration: Aug 27 2001 → Aug 31 2001 |

### 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 | 2136 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001 |
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Country | Czech Republic |

City | Marianske Lazne |

Period | 8/27/01 → 8/31/01 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings*(Vol. 2136, pp. 585-597). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2136). Springer Verlag.