A note on succinct representations of graphs
Information and Control
The complexity of reliability computations in planar and acyclic graphs
SIAM Journal on Computing
The complexity of congestion-1 embedding in a hypercube
Journal of Algorithms
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
A very hard log-space counting class
Theoretical Computer Science - Special issue on structure in complexity theory
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
A linear-time algorithm for drawing a planar graph on a grid
Information Processing Letters
The Complexity of Planar Counting Problems
SIAM Journal on Computing
Tally NP sets and easy census functions
Information and Computation
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
The Complexity of Counting in Sparse, Regular, and Planar Graphs
SIAM Journal on Computing
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Some connections between nonuniform and uniform complexity classes
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
SIGACT News complexity theory column 41
ACM SIGACT News
Completeness for parity problems
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
#3-Regular bipartite planar vertex cover is #p-complete
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
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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.