Abstract:
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.

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 #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.

Abstract:
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.