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

N1 - Funding Information:
Supported in part by NSF grants CCR-9701911, CCR-9725021, DUE-9980943, INT-9726724, an NIH/NIA grant RO1-AG18231 and a DARPA grant F30602-98-2-013.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2001.

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.

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U2 - 10.1007/3-540-44683-4_51

DO - 10.1007/3-540-44683-4_51

M3 - Conference contribution

AN - SCOPUS:84974693472

SN - 9783540446835

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

A2 - Sgall, Jiri

A2 - Pultr, Ales

A2 - Kolman, Petr

PB - Springer Verlag

T2 - 26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001

Y2 - 27 August 2001 through 31 August 2001

ER -