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 Scopus citations

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
EditorsJiri Sgall, Ales Pultr, Petr Kolman
PublisherSpringer Verlag
Pages585-597
Number of pages13
ISBN (Print)9783540446835
DOIs
StatePublished - 2001
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)0302-9743
ISSN (Electronic)1611-3349

Other

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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    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 J. Sgall, A. Pultr, & P. Kolman (Eds.), Mathematical Foundations of Computer Science 2001 - 26th International Symposium, MFCS 2001, Proceedings (pp. 585-597). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2136). Springer Verlag. https://doi.org/10.1007/3-540-44683-4_51