### Abstract

Valiant (SIAM J. Comput. 8 (1979) 410-421) showed that the problem of computing the number of simple s-t paths in graphs is #P-complete both in the case of directed graphs and in the case of undirected graphs. Welsh (Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, 1993, p. 17) asked whether the problem of computing the number of self-avoiding walks of a given length in the complete two-dimensional grid is complete for #P_{1}, the tally-version of #P. This paper offers a partial answer to the question of Welsh: it is #P-complete to compute the number of self-avoiding walks of a given length in a subgraph of a two-dimensional grid. Several variations of the problem are also studied and shown to be #P-complete. This paper also studies the problem of computing the number of self-avoiding walks in a subgraph of a hypercube. Similar completeness results are shown for the problem. By scaling the computation time to exponential, it is shown that computing the number of self-avoiding walks in hypercubes is a complete problem for #EXP in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes. Finally, this paper studies the complexity of testing whether a given word over the four-letter alphabet {U,D,L,R} represents a self-avoiding walk in a two-dimensional grid. A linear-space lower bound is shown for nondeterministic Turing machines with a 1-way input head to make this test.

Original language | English (US) |
---|---|

Pages (from-to) | 129-156 |

Number of pages | 28 |

Journal | Theoretical Computer Science |

Volume | 304 |

Issue number | 1-3 |

DOIs | |

State | Published - Jul 28 2003 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Theoretical Computer Science*,

*304*(1-3), 129-156. https://doi.org/10.1016/S0304-3975(03)00080-X

**The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes.** / Liśkiewicz, Maciej; Ogihara, Mitsunori; Toda, Seinosuke.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 304, no. 1-3, pp. 129-156. https://doi.org/10.1016/S0304-3975(03)00080-X

}

TY - JOUR

T1 - The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes

AU - Liśkiewicz, Maciej

AU - Ogihara, Mitsunori

AU - Toda, Seinosuke

PY - 2003/7/28

Y1 - 2003/7/28

N2 - Valiant (SIAM J. Comput. 8 (1979) 410-421) showed that the problem of computing the number of simple s-t paths in graphs is #P-complete both in the case of directed graphs and in the case of undirected graphs. Welsh (Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, 1993, p. 17) asked whether the problem of computing the number of self-avoiding walks of a given length in the complete two-dimensional grid is complete for #P1, the tally-version of #P. This paper offers a partial answer to the question of Welsh: it is #P-complete to compute the number of self-avoiding walks of a given length in a subgraph of a two-dimensional grid. Several variations of the problem are also studied and shown to be #P-complete. This paper also studies the problem of computing the number of self-avoiding walks in a subgraph of a hypercube. Similar completeness results are shown for the problem. By scaling the computation time to exponential, it is shown that computing the number of self-avoiding walks in hypercubes is a complete problem for #EXP in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes. Finally, this paper studies the complexity of testing whether a given word over the four-letter alphabet {U,D,L,R} represents a self-avoiding walk in a two-dimensional grid. A linear-space lower bound is shown for nondeterministic Turing machines with a 1-way input head to make this test.

AB - Valiant (SIAM J. Comput. 8 (1979) 410-421) showed that the problem of computing the number of simple s-t paths in graphs is #P-complete both in the case of directed graphs and in the case of undirected graphs. Welsh (Complexity: Knots, Colourings and Counting, Cambridge University Press, Cambridge, 1993, p. 17) asked whether the problem of computing the number of self-avoiding walks of a given length in the complete two-dimensional grid is complete for #P1, the tally-version of #P. This paper offers a partial answer to the question of Welsh: it is #P-complete to compute the number of self-avoiding walks of a given length in a subgraph of a two-dimensional grid. Several variations of the problem are also studied and shown to be #P-complete. This paper also studies the problem of computing the number of self-avoiding walks in a subgraph of a hypercube. Similar completeness results are shown for the problem. By scaling the computation time to exponential, it is shown that computing the number of self-avoiding walks in hypercubes is a complete problem for #EXP in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes. Finally, this paper studies the complexity of testing whether a given word over the four-letter alphabet {U,D,L,R} represents a self-avoiding walk in a two-dimensional grid. A linear-space lower bound is shown for nondeterministic Turing machines with a 1-way input head to make this test.

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U2 - 10.1016/S0304-3975(03)00080-X

DO - 10.1016/S0304-3975(03)00080-X

M3 - Article

VL - 304

SP - 129

EP - 156

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1-3

ER -