The relative complexity of approximate counting problems

Algorithmica (New York)

Volumn 38, Issue 3, 2003, Pages 471-500

  Dyer, Martin ^a   Goldberg, Leslie Ann ^b   Greenhill, Catherine ^c   Jerrum, Mark ^d  

^a UNIVERSITY OF LEEDS   (United Kingdom)

^b UNIVERSITY OF WARWICK   (United Kingdom)

^c UNIVERSITY OF MELBOURNE   (Australia)

^d UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Approximate counting; Computational complexity

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; INTEGER PROGRAMMING; MATHEMATICAL MODELS; POLYNOMIALS; PROBLEM SOLVING; TURING MACHINES;

APPROXIMATE COUNTING; APPROXIMATION ALGORITHMS; INTERMEDIATE COMPLEXITY; RANDOMIZED APPROXIMATION SCHEME (RAS);

APPROXIMATION THEORY;

EID: 21144434561     PISSN: 01784617     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00453-003-1073-y     Document Type: Article

References (23)

1. G. Brightwell, O. Häggström and P. Winkler, Nonmonotonic behavior in hard-core and Widom-Rowlinson models. Journal of Statistical Physics 94 (1999). 415-135.
2. G.R. Brightwell and P. Winkler, Graph homomorphisms and phase transitions, CDAM Research Report, LSE-CDAM-97-01, Centre for Discrete and Applicable Mathematics, London School of Economics, October 1997.
3. R.L. Brooks, On colouring the nodes of a network, Proceedings of the Cambridge Philosophical Society 37 (1941), 194-197.
4. M. Dyer, A. Frieze and M. Jerrum, On counting independent sets in sparse graphs, Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS '99), IEEE Computer Society Press, Los Alamitos, CA, 1999, pp. 210-217.
5. M. Dyer and C. Greenhill, The complexity of counting graph homomorphisms, Random Structures and Algorithms, Vol. 7, pp. 260-289, 2000.
6. (A preliminary version appeared in Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (2000), pp. 246-255.)
7. R. Fagin, Generalized first-order spectra and polynomial time recognisable sets. In Complexity of Computation (R. Karp, ed.), SIAM-AMS Proceedings, Vol. 7, American Mathematical Society, Providence, RI, 1974, pp. 43-73.
8. M.R. Garey, D.S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theoretical Computer Science 1 (1976), 237-267.
9. O. Goldreich, Introduction to Complexity Theory, Lecture Notes Series of the Electronic Colloquium on Computational Complexity, 1999, Chapter 10: "#P and approximating it." http://www.eccc.uni-trier.de/eccc-local/ECCC-LectureNotes/
10. M. Jerrum and A. Sinclair, Polynomial-time approximation algorithms for the Ising model, SIAM Journal on Computing 22 (1993), 1087-1116.
11. M. Jerrum and A. Sinclair, The Markov chain Monte Carlo method: an approach to approximate counting and integration. In Approximation Algorithms for NP-Hard Problems (D. Hochbaum, ed.), PWS, Boston, MA, 1996, pp. 482-520.
12. M.R. Jerrum, L.G. Valiant and V.V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science 43 (1986), 169-188.
13. R.M. Karp, M. Luby and N. Madras, Monte-Carlo approximation algorithms for enumeration problems, Journal of Algorithms 10 (1989), 429-448.
14. P. W. Kasteleyn, Graph theory and crystal physics. In Graph Theory and Statistical Physics (F. Harary, ed.), Academic Press, New York, 1967, pp. 43-110.
15. J.S. Provan and M.O. Ball, The complexity of counting cuts and of computing the probability that a graph is connected, SIAM Journal on Computing 12 (1983), 777-788.
16. D. Roth, On the hardness of approximate reasoning. Artificial Intelligence Journal 82 ( 1996), 273-302.
17. [16J S. Saluja, K.V. Subrahmanyam and M.N. Thakur, Descriptive complexity of #P functions, Journal of Computer and Systems Sciences 50 (1995), 493-505.
18. J. Simon, On the difference between one and many (preliminary version), Proceedings of the 4th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 52, Springer-Verlag, Berlin, 1977, pp. 480-491.
19. A. Sinclair, Algorithms for Random Generation and Counting: A Markov Chain Approach, Progress in Theoretical Computer Science, Birkhäuser, Boston, MA. 1993.
20. L. Stockmeyer, The complexity of approximate counting (preliminary version), Proceedings of the 15th ACM Symposium on Theory of Computing (STOC '83), ACM, New York 1983, pp. 118-126.
21. S. Toda, PP is as hard as the polynomial-time hierarchy, SIAM Journal on Computing 20 (1991), 865-877.
22. L.G. Valiant and V.V. Vazirani, NP is as easy as detecting unique solutions. Theoretical Computer Science 47 (1986), 85-93.
23. D. Zuckerman, On unapproximable versions of NP-complete problems. SIAM Journal on Computing 25 (1996), 1293-1304.