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

Maciej Liśkiewicz, Mitsunori Ogihara, Seinosuke Toda

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 #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.

Original languageEnglish (US)
Pages (from-to)129-156
Number of pages28
JournalTheoretical Computer Science
Volume304
Issue number1-3
DOIs
StatePublished - Jul 28 2003
Externally publishedYes

Fingerprint

Self-avoiding Walk
Hypercube
Subgraph
Counting
Grid
Turing machines
Computing
Directed graphs
Tally
Boolean Circuits
Networks (circuits)
Turing Machine
Testing
Linear Space
Undirected Graph
Directed Graph
Knot
Completeness
Scaling
Lower bound

ASJC Scopus subject areas

  • Computational Theory and Mathematics

Cite this

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

In: Theoretical Computer Science, Vol. 304, No. 1-3, 28.07.2003, p. 129-156.

Research output: Contribution to journalArticle

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