The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs

Mitsunori Ogihara, Seinosuke Toda

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

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 languageEnglish (US)
Title of host publicationMathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings
PublisherSpringer Verlag
Pages585-597
Number of pages13
Volume2136
ISBN (Print)9783540446835
StatePublished - 2001
Externally publishedYes
Event26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001 - Marianske Lazne, Czech Republic
Duration: Aug 27 2001Aug 31 2001

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2136
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001
CountryCzech Republic
CityMarianske Lazne
Period8/27/018/31/01

Fingerprint

Grid Graph
Self-avoiding Walk
Hypercube
Turing
Polynomials
Computing
Polynomial time
Graph in graph theory
Counting
Embedded Graph
Directed graphs
Tally
Grid
Boolean Circuits
Vertex of a graph
Undirected Graph
Walk
Directed Graph
Networks (circuits)
Completeness

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Ogihara, M., & Toda, S. (2001). The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs. In 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.

The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs. / Ogihara, Mitsunori; Toda, Seinosuke.

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Ogihara, M & Toda, S 2001, The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs. in Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings. vol. 2136, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2136, Springer Verlag, pp. 585-597, 26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001, Marianske Lazne, Czech Republic, 8/27/01.
Ogihara M, Toda S. The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs. In Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings. Vol. 2136. Springer Verlag. 2001. p. 585-597. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
Ogihara, Mitsunori ; Toda, Seinosuke. / The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs. Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings. Vol. 2136 Springer Verlag, 2001. pp. 585-597 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{3b4dc97c362c4b748345f8f4925269e3,
title = "The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs",
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.",
author = "Mitsunori Ogihara and Seinosuke Toda",
year = "2001",
language = "English (US)",
isbn = "9783540446835",
volume = "2136",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
publisher = "Springer Verlag",
pages = "585--597",
booktitle = "Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings",
address = "Germany",

}

TY - GEN

T1 - The complexity of computing the number of self-avoiding walks in two-dimensional grid graphs and in hypercube graphs

AU - Ogihara, Mitsunori

AU - Toda, Seinosuke

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84974693472&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84974693472&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84974693472

SN - 9783540446835

VL - 2136

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 585

EP - 597

BT - Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings

PB - Springer Verlag

ER -