## Computational Pólya theory (1995)

Venue: | In Surveys in combinatorics |

Citations: | 7 - 1 self |

### BibTeX

@INPROCEEDINGS{Jerrum95computationalpólya,

author = {Mark Jerrum},

title = {Computational Pólya theory},

booktitle = {In Surveys in combinatorics},

year = {1995},

pages = {103--118},

publisher = {University Press}

}

### OpenURL

### Abstract

A permutation group G of degree n has a natural induced action on words of length n over a finite alphabet \Sigma , in which the image x of x under permutation g 2 G is obtained by permuting the positions of symbols in x according to g. The key result in "P'olya theory" is that the number of orbits of this action is given by an evaluation of the cycleindex polynomial PG (z 1 ; : : : ; z n ) of G at the point z 1 = \Delta \Delta \Delta = z n = j\Sigma j.

### Citations

11502 | Computers and Intractability, A Guide to the Theory of NPCompleteness - Garey, Johnson - 1979 |

528 |
The complexity of computing the permanent
- Valiant
- 1979
(Show Context)
Citation Context ...olynomial time if there exists a Turing machine that computes f(ff 1 ; : : : ; ff m ) in time polynomial in jff 1 j + \Delta \Delta \Delta + jff m j. The complexity class #P was introduced by Valiant =-=[22, 23]-=- as a counting analogue to the more familiar class NP of decision problems. It may be defined in several equivalent ways, of which the following is particularly well suited to our purpose. A function ... |

446 |
The Complexity of Enumeration and Reliability Problems
- Valiant
- 1979
(Show Context)
Citation Context ...olynomial time if there exists a Turing machine that computes f(ff 1 ; : : : ; ff m ) in time polynomial in jff 1 j + \Delta \Delta \Delta + jff m j. The complexity class #P was introduced by Valiant =-=[22, 23]-=- as a counting analogue to the more familiar class NP of decision problems. It may be defined in several equivalent ways, of which the following is particularly well suited to our purpose. A function ... |

319 |
Graphical Enumeration
- Harary, Palmer
- 1973
(Show Context)
Citation Context ...z 1 = \Delta \Delta \Delta = z n = k. For many important choices for G, this computation is feasible and leads to results concerning the number of unlabelled combinatorial structures of various kinds =-=[11]-=-. In this article we study the problem of computing PG for an arbitrary permutation group G and at an arbitrary point. 1.2 Algorithmic preliminaries It is clear that the cycle-index polynomial of a pe... |

306 |
Group representations in probability and statistics
- Diaconis
- 1988
(Show Context)
Citation Context ... M 0 match the sampling distributions specified in questions (b) and (c). Although the topic of random walks on groups has received much attention (see for example the work of Aldous [1] and Diaconis =-=[5]-=-), previous authors have been concerned with walks which converge to a uniform distribution. The novel aspect of the current investigation is that the stationary distribution is required to be highly ... |

298 | Approximating the permanent
- Jerrum, Sinclair
- 1989
(Show Context)
Citation Context ...rkov chain. This technique has proved fruitful on a number of occasions in recent years; previous applications include an algorithm of Jerrum and Sinclair for estimating the permanent of a 0,1-matrix =-=[15]-=- and one of Dyer, Frieze, and Kannan for estimating the volume of a convex body in ndimensional space [6]. In this instance we wish to construct a Markov chain M = M(G; \Sigma ) whose state space is \... |

283 |
Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science 43
- Jerrum, Valiant, et al.
- 1986
(Show Context)
Citation Context ...able) such that Pr i (1 \Gamma ")P G (k; : : : ; k)sYs(1 + ")P G (k; : : : ; k) j 3 4 ; and, moreover, does so within time poly(n; " \Gamma1 )? (b) Is there a polynomial-time almost uni=-=form sampler 4 [16] for -=-the orbits of \Sigma n under the action of G? That is to say, is there a randomised algorithm that takes as input a group G and " ? 0, and produces as output a word Y 2 \Sigma n (a random variabl... |

231 |
Random walks on finite groups and rapidly mixing Markov chains
- Aldous
- 1983
(Show Context)
Citation Context ...at has vertex bipartition (\Sigma n ; G) and edge set f(ff; g) : ff g = ffg. It is clear that the Markov chain M may be viewed as sampling the random walk on B after every even step. Lets: \Sigma n ! =-=[0; 1]-=- denote the stationary distribution of M . Then (x) is proportional to the degree of vertex ff in the graph B, which is jG ff j. It is an elementary grouptheoretic fact that jG ff j \Theta jff G j = j... |

176 | Improved bounds for mixing rates of Markov chains and multicommodity flow
- Sinclair
- 1992
(Show Context)
Citation Context ...tion between mixing rate and the expansion properties of the Markov chain viewed as a graph. Sinclair has provided a useful survey of these techniques, in addition to presenting some sharpened bounds =-=[20]-=-. Nevertheless, proofs of rapid mixing still tend to be technically involved. Since no counterexamples have been identified, it remains a possibility that for any fixed alphabet \Sigma , the Markov ch... |

152 |
Isomorphism of graphs of bounded valence can be tested in polynomial time
- Luks
- 1982
(Show Context)
Citation Context ...ps G for which step (1) does have an efficient implementation. Luks has shown that p-groups --- groups in which every element has order a power of p for some prime p --- is an example of such a class =-=[18]-=-. Returning to the Markov chain itself, we note immediately that M is ergodic, since every state can be reached from every other in a single transition, by selecting the identity permutation in step (... |

149 | A random polynomial time algorithm for approximating the volume of convex bodies
- Dyer, Frieze, et al.
- 1991
(Show Context)
Citation Context ...ions include an algorithm of Jerrum and Sinclair for estimating the permanent of a 0,1-matrix [15] and one of Dyer, Frieze, and Kannan for estimating the volume of a convex body in ndimensional space =-=[6]-=-. In this instance we wish to construct a Markov chain M = M(G; \Sigma ) whose state space is \Sigma n , and whose stationary distribution assigns equal probability to each orbit. In fact, we shall ai... |

124 |
Monte-Carlo algorithms for enumeration and reliability problems
- Karp, Luby
- 1983
(Show Context)
Citation Context ... one of a group of three related questions that may be formalised as follows. (As before \Sigma is a finite alphabet of cardinality k.) (a) Is there a fully polynomial randomised approximation scheme =-=[17] (fpr-=-as) for estimating PG (k; : : : ; k)? That is to say, is there a randomised algorithm that takes as input a group G and " ? 0, and produces as output a number Y (a random variable) such that Pr i... |

55 |
Computational methods in the study of permutation groups
- Sims
- 1970
(Show Context)
Citation Context ...her hand, as we shall see presently, every permutation group G of degree n has a simple and compact (in terms of n) encoding that allows many questions about G to answered efficiently. Following Sims =-=[19]-=-, for 1sisn let G i = fg 2 G : j g = j; for all 0sj ! ig; be the subgroup of G stabilising [i] pointwise, and let G 0 = G. Then G = G 0sG 1s\Delta \Delta \DeltasG n = f1g is a sequence of subgroups of... |

31 |
Efficient Algorithms for Listing Combinatorial Structures
- Goldberg
- 1993
(Show Context)
Citation Context ...e notion of efficient solvability. It would be surprising if all NP-complete problems were efficiently solvable in the RP sense, and so it is conjectured that RP is strictly contained in NP. Goldberg =-=[9, 10]-=- has shown the following. Theorem 4 Let (a i ) be a sequence of non-negative rationals, such that there exists i with a i ? a i 1 . There can be no fpras for CycleIndex(a i ), unless RP = NP. The abov... |

27 |
Polynomial time algorithms for permutation groups
- Furst, Hopcroft, et al.
- 1980
(Show Context)
Citation Context ... strong generating set for G 1 . Making realistic assumptions about the model of computation, the decision procedure just sketched can be implemented to run in time O(n 2 ). Furst, Hopcroft, and Luks =-=[7]-=-, who were the first to analyse Sims' data structure from a complexity-theoretic viewpoint, showed that a strong generating set may be computed in time O(n 6 ) from an arbitrary (small) set of generat... |

27 |
A compact representation for permutation groups
- JERRUM
- 1986
(Show Context)
Citation Context ...showed that a strong generating set may be computed in time O(n 6 ) from an arbitrary (small) set of generators for G. This bound on time-complexity was subsequently improved to O(n 5 ) by the author =-=[12]-=- using elementary techniques, and to e O(n 4 ) by Babai, Luks, and Seress [2], using non-elementary techniques relying on the classification of finite simple groups. 2 2 The e O() notation hides not m... |

20 | Randomised algorithms for counting and generating combinatorial structures - Sinclair - 1993 |

13 |
Uniform sampling modulo a group of symmetries using Markov chain simulation
- JERRUM
- 1993
(Show Context)
Citation Context ...er resolving question (a) in the affirmative would immediately settle either of the others. The two known entailments are described in the following proposition, whose (routine) proof may be found in =-=[13, 14]-=-. Proposition 3 An affirmative answer to question (c) would entail affirmative answers to (a) and (b). I feel that questions (a)--(c) are quite significant, and they are all unresolved. It is worth co... |

11 |
Ákos Seress. Fast management of permutation groups
- Babai, Luks
- 1997
(Show Context)
Citation Context ...bitrary (small) set of generators for G. This bound on time-complexity was subsequently improved to O(n 5 ) by the author [12] using elementary techniques, and to e O(n 4 ) by Babai, Luks, and Seress =-=[2]-=-, using non-elementary techniques relying on the classification of finite simple groups. 2 2 The e O() notation hides not merely constants, but also arbitrary powers of log n. 4 Mark Jerrum 1.3 Comple... |

8 |
Automating Polya theory: The computational complexity of the cycle index polynomial
- Goldberg
- 1993
(Show Context)
Citation Context ...sult of Goldberg. Theorem 2 Let (a i ) be a sequence of non-negative rational numbers satisfyingsa i 6= a i 1 ; 0, for some i. Then CycleIndex(a i ) is #P-hard. A proof may be found in [9, p. 150] or =-=[10]-=-. Note that we cannot claim #Pcompleteness here; however, if we assume in addition that (a i ) is a sequence of natural numbers computable in time polynomial in i, then CycleIndex(a i ) is in #P, and ... |

3 |
de Bruijn, Pólya’s Theory of Counting, Applied Combinatorial Mathematics, chapter 5
- G
- 1964
(Show Context)
Citation Context ...using a like construction. What has just been described is the setting for P'olya theory, the key result of which is an expression for the number of orbits in terms of the cycle-index polynomial of G =-=[4]-=-. This polynomial, in the variables z 1 ; z 2 ; : : : ; z n , is defined to be PG (z 1 ; : : : ; z n ) = jGj \Gamma1 X g2G z 1 c 1 (g) : : : z n cn (g) ; (1) where c i (g) denotes the number of cycles... |