zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Approximate counting, uniform generation and rapidly mixing Markov chains. (English) Zbl 0668.05060
The paper studies effective approximate solutions to combinatorial counting and uniform generation problems. Using a technique based on the simulation of ergodic Markov chains, it is shown that, for self-reducible structures, almost uniform generation is possible in polynomial time provided only that randomized approximate counting to within some arbitrary polynomial factor is possible in polynomial time. It follows that, for self-reducible structures, polynomial time randomized algorithms for counting to within factors of the form $(1+n\sp{-\beta})$ are available either for all $\beta\in {\bbfR}$ or for no $\beta\in {\bbfR}$. A substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic Markov chains, and a simple but powerful characterization of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established. Finally, the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomized approximate counting algorithms for these graphs with very good asymptotic behaviour.

05C99Graph theory
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
Full Text: DOI
[1] Aldous, D.: Random walks on finite groups and rapidly mixing Markov chains. Lecture notes in mathematics 986, 243-297 (1981)
[2] Aldous, D.: On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Prob. engrg. Inform. sci. 1, 33-46 (1987) · Zbl 1133.60327
[3] Aldous, D.; Diaconis, P.: Shuffling cards and stopping times. Amer. math. Monthly 93, 333-348 (1986) · Zbl 0603.60006
[4] Aldous, D.; Diaconis, P.: Strong uniform times and finite random walks. Adv. in appl. Math. 8, 69-97 (1987) · Zbl 0631.60065
[5] Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83-96 (1986) · Zbl 0661.05053
[6] Alon, N.; Milman, V. D.: {$\lambda$}1, isoperimetric inequalities for graphs and superconcentrators. J. combin. Theory ser. B 38, 73-88 (1985) · Zbl 0549.05051
[7] Bach, E.: How to generate random integers with known factorisation. Proceedings 15th ACM symposium on theory of computing, 184-188 (1983)
[8] Binder, K.: Monte Carlo investigations of phase transitions and critical phenomena. Phase transitions and critical phenomena 5b, 1-105 (1976)
[9] Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1, 311-316 (1980) · Zbl 0457.05038
[10] Bollobás, B.: Random graphs. (1985) · Zbl 0567.05042
[11] Broder, A. Z.: How hard is it to marry at random? (On the approximation of the permanent). Proceedings, 18th ACM symposium on theory of computing, 50-58 (1986)
[12] Cai, J. -Y.; Hemachandra, L. A.: Exact counting is as easy as approximate counting. (1986)
[13] Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis, 195-199 (1970) · Zbl 0212.44903
[14] Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. amer. Math. soc. 284, 787-794 (1984) · Zbl 0512.39001
[15] Feller, W.: 3rd ed. An introduction to probability theory and its applications. An introduction to probability theory and its applications (1968) · Zbl 0155.23101
[16] Garey, M. R.; Johnson, D. S.: Computers and intractability: A guide to the theory of NP-completeness. (1979) · Zbl 0411.68039
[17] Gill, J.: Computational complexity of probabilistic Turing machines. SIAM J. Comput. 6, 675-695 (1977) · Zbl 0366.02024
[18] Guénoche, A.: Random spanning tree. J. algorithms 4, 214-220 (1983) · Zbl 0521.68073
[19] Jerrum, M. R.; Sinclair, A. J.: Conductance and the rapid mixing property for Markov chains: the approximation of the permanent resolved. Proceedings, 20th ACM symposium on theory of computing, 235-244 (1988)
[20] Jerrum, M. R.; Valiant, L. G.; Vazirani, V. V.: Random generation of combinatorial structures from a uniform distribution. Theoret. comput. Sci. 43, 169-188 (1986) · Zbl 0597.68056
[21] Karp, R. M.; Luby, M.: Monte-Carlo algorithms for enumeration and reliability problems. Proceedings, 24th IEEE symposium on foundations of computer science, 56-64 (1983) · Zbl 0596.90033
[22] Keilson, J.: Markov chain models--rarity and exponentiality. (1979) · Zbl 0411.60068
[23] Kirkpatrick, S.; Gellatt, C. D.; Vecchi, M. P.: Optimisation by simulated annealing. Science 220, 671-680 (1983) · Zbl 1225.90162
[24] Kolmogorov, A. N.; Fomin, S. V.: Introductory real analysis. (1970) · Zbl 0213.07305
[25] Lawler, G. F.; Sokal, A. D.: Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. Trans. amer. Math. soc. 309, 557-580 (1988) · Zbl 0716.60073
[26] Mckay, B. D.: Asymptotics for symmetric 0--1 matrices with prescribed row sums. Ars combin. 19A, 15-25 (1985) · Zbl 0568.05032
[27] Nijenhuis, A.; Wilf, H. S.: 2nd ed. Combinatorial algorithms. Combinatorial algorithms (1978)
[28] Schnorr, C. P.: Optimal algorithms for self-reducible problems. Proceedings, 3rd international colloquium on automata, languages and programming, 322-337 (1976)
[29] Seneta, E.: 2nd ed. Non-negative matrices and Markov chains. Non-negative matrices and Markov chains (1981) · Zbl 0471.60001
[30] Sinclair, A. J.: Randomised algorithms for counting and generating combinatorial structures. Ph.d. thesis (1988)
[31] Stockmeyer, L.: The polynomial-time hierarchy. Theoret. comput. Sci. 3, 1-22 (1977) · Zbl 0353.02024
[32] Stockmeyer, L.: The complexity of approximate counting. Proceedings, 15th ACM symposium on theory of computing, 118-126 (1983)
[33] Tinhofer, G.: On the generation of random graphs with given properties and known distribution. Appl. comput. Sci., ber. Prakt. inf. 13, 265-297 (1979) · Zbl 0421.05064
[34] Valiant, L. G.: The complexity of computing the permanent. Theoret. comput. Sci. 8, 189-201 (1979) · Zbl 0415.68008
[35] Valiant, L. G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410-421 (1979) · Zbl 0419.68082
[36] Wilf, H. S.: The uniform selection of free trees. J. algorithms 2, 204-207 (1981)
[37] Wormald, N. C.: Generating random regular graphs. J. algorithms 5, 247-280 (1984) · Zbl 0543.68048
[38] Wormald, N. C.: Generating random unlabelled graphs. SIAM J. Comput. 16, 717-727 (1987) · Zbl 0634.68067