Valiant (SIAM Journal on Computing 8, pages 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. Valiant then asked whether the self-avoiding walk
problem on the two-dimensional grid, the problem of computing the
number of self-avoiding walks of a given length in the two-dimensional
grid is complete for #P_1, the tally-version of #P. This paper offers
a partial answer to the question of Valiant. It is shown that computing
the number of self-avoiding walks of a given length in the
two-dimensional grid graph is #P-complete. The paper also studies
several variations of the prolem and shows that all of them are
#P-complete.

This paper also studies the problem of computing the number of
self-avoiding walks in graphs embedded in a hypercube. Similar
completeness results are shown for hypercube graphs. By scaling the
computation time to exponential, it is shown that computing the number
fo self-avoiding walks in the hypercubes is complete for #EXP in the
case when a hypercube graph 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 } is a self-avoiding
walk. A linear-space lower bound is shown for nondeterministic Turing
machines with a one-way input head to recognize self-avoiding walks.

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 #P_1, 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
two-dimensional 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
self-avoiding 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 oneu
that accepts its edges.

Finally, this paper studies the complexity of testing whether a given
word over the four letter alphabet { U, L, D, R } is a self-avoiding
walk. It shows a linear-space lower bound for nondeterminstic Turing
machines with a one-way input head.

Valiant (SIAM Journal on Computing 8, pages 410--421) showed that the
roblem 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
two-dimensional grid graphs (graphs embedded in the two-dimensional
grid) is polynomial-time one-Turing complete for #P.