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

Theoretical Computer Science

Volumn 304, Issue 1-3, 2003, Pages 129-156

  Liśkiewicz, Maciej ^a   Ogihara, Mitsunori ^b   Toda, Seinosuke ^c  

^a UNIVERSITY OF LÜBECK   (Germany)

^b University of Rochester   (United States)

^c NIHON UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; GRAPH THEORY; PROBLEM SOLVING;

HYPERCUBES;

COMPUTER SCIENCE;

EID: 0037811208     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(03)00080-X     Document Type: Article

References (28)

1. Allender E., Ogihara M. Relationships among PL, #L , and the determinant RAIRO - Theoret. Inform. Appl. 30:1996;1-21.
2. Álvarez C., Jenner B. A very hard Log-space counting classes. Theoret. Comput. Sci. 107:1993;3-30.
3. Barahona F. On the computational complexity of Ising spin glass models. J. Phys. A. 15(3):1982;3241-3245.
4. Chrobak M., Payne T.H. A linear time algorithm for drawing a planar graph on a grid. Inform. Process. Lett. 54(4):1995;241-246.
5. Conway A.R., Guttmann A.J. Square lattice self-avoiding walks and corrections-to-scaling. Phys. Rev. Lett. 77(26):1996;5284-5287.
6. Garey M., Johnson D., Tarjan E. The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4):1976;704-714.
7. Goldsmith J., Ogihara M., Rothe J. Tally NP sets and easy census functions. Inform. Comput. 158:2000;29-52.
8. Hammersley J.M., Morton K.W. Poor man's Monte Carlo. J. Roy. Statist. Soc. B. 16:1954;23-38.
9. Hammersley J.M., Welsh D.J.A. Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford. 13(2):1962;108-110.
10. Hunt III H.B., Marathe M.V., Radhakrishnan V., Stearns R.E. The complexity of planar counting problems. SIAM J. Comput. 27(4):1998;1142-1167.
11. S. Istrail, Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intractability for the partition function of the Ising model across non-planar surfaces, in: Proc. 31st Symp. on Theory of Computing, ACM Press, New York, 2000, pp. 87-96.
12. R.M. Karp, R.J. Lipton, Some connections between nonuniform and uniform complexity classes, in: Proc. 12th Symp. on Theory of Computing, ACM Press, New York, 1980. pp. 302-309.
13. Kim Y.M., Lai T.-H. The complexity of congestion-1 embedding in a hypercube. J. Algorithms. 12:1991;246-280.
14. Madras N., Slade G. The Self-Avoiding Walk. 1993;Birkhäuser, Boston, MA.
15. Matsubayashi A., Ueno S. Small congestion embedding of graphs into hypercubes. Networks. 33:1999;71-77.
16. Papadimitriou C. Computational Complexity. 1994;Addison-Wesley, Reading, MA.
17. Papadimitriou C., Yannakakis M. A note on succinct representations of graphs. Inform. Control. 71(3):1986;181-185.
18. d , Electron. J. Combinatorics 7 (1) (2000) #R21. http://www.combinatorics.org/Volume_7/v7i1toc.html.
19. Provan J.S. The complexity of reliability computations in planar and acyclic graphs. SIAM J. Comput. 15(3):1986;694-702.
20. Randall D., Sinclair A. Self-testing algorithms for self-avoiding walks. J. Math. Phys. 41(3):2000;1570-1584.
21. Schaefer T. The complexity of satisfiability problems. Proc. 10th Symp. on Theory of Computing. 1978;216-226 ACM Press, New York.
22. Sloane N.J.A., Plouffe S. The Encyclopedia of Integer Sequences. 1995;Academic Press, San Diego, CA.
23. Toda S. PP is as hard as the polynomial time hierarchy. SIAM J. Comput. 20(5):1991;865-877.
24. Vadhan S.P. The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2):2001;398-427.
25. Valiant L. The complexity of computing the permanent. Theoret. Comput. Sci. 8(2):1979;189-201.
26. Valiant L. The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3):1979;410-421.
27. Welsh D. Complexity: Knots, Colourings and Counting. 1993;Cambridge University Press, Cambridge.
28. Williams L. Enumerating up-side self-avoiding walks on integer lattices. Electron. J. Combinatorics. 3(1):1996;. Research Paper 31.