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

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. © 2002 Elsevier B.V. All Rights Reserved.