scholarly journals An Analysis of Phase Transition in NK Landscapes

2002 ◽  
Vol 17 ◽  
pp. 309-332 ◽  
Author(s):  
Y. Gao ◽  
J. Culberson
Keyword(s):  
Phase Transition ◽  
Probability Model ◽  
Fixed Ratio ◽  
Upper Bounds ◽  
Ratio Model ◽  
Problem Size ◽  
Threshold Phenomena ◽  
Uniform Probability ◽  
Random Instances ◽  
Nk Landscapes

In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

scholarly journals Tail Bounds for the Stable Marriage of Poisson and Lebesgue

2009 ◽  
Vol 61 (6) ◽  
pp. 1279-1299 ◽  
Author(s):  
Christopher Hoffman ◽  
Alexander E. Holroyd ◽  
Yuval Peres
Keyword(s):  
Phase Transition ◽  
Lower Bounds ◽  
Power Law ◽  
Critical Case ◽  
Voronoi Tessellation ◽  
Upper Bounds ◽  
Stable Marriage ◽  
Marriage Problem ◽  
Stable Allocation ◽  
Typical Site

Abstract Let 𝚵 be a discrete set in ℝd. Call the elements of 𝚵 centers. The well-known Voronoi tessellation partitions ℝd into polyhedral regions (of varying volumes) by allocating each site of ℝd to the closest center. Here we study allocations of ℝd to 𝚵 in which each center attempts to claima region of equal volume α.We focus on the case where 𝚵 arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance X from a typical site to its allocated center in the stable allocation.The model exhibits a phase transition in the appetite α. In the critical case α = 1 we prove a power law upper bound on X in dimension d = 1. (Power law lower bounds were proved earlier for all d). In the non-critical cases α < 1 and α > 1 we prove exponential upper bounds on X.

Drift and Scaling in Estimation of Distribution Algorithms

2005 ◽  
Vol 13 (1) ◽  
pp. 99-123 ◽  
Author(s):  
J. L. Shapiro
Keyword(s):  
Population Size ◽  
Probability Model ◽  
Search Space ◽  
System Size ◽  
Estimation Of Distribution Algorithms ◽  
Square Root ◽  
Problem Size ◽  
Probability Models ◽  
Estimation Of Distribution ◽  
Distribution Algorithms

This paper considers a phenomenon in Estimation of Distribution Algorithms (EDA) analogous to drift in population genetic dynamics. Finite population sampling in selection results in fluctuations which get reinforced when the probability model is updated. As a consequence, any probability model which can generate only a single set of values with probability 1 can be an attractive fixed point of the algorithm. To avoid this, parameters of the algorithm must scale with the system size in strongly problem-dependent ways, or the algorithm must be modified. This phenomenon is shown to hold for general EDAs as a consequence of the lack of ergodicity and irreducibility of the Markov chain on the state of probability models. It is illustrated in the case of UMDA, in which it is shown that the global optimum is only found if the population size is sufficiently large. For the needle-in-a haystack problem, the population size must scale as the square-root of the size of the search space. For the one-max problem, the population size must scale as the square-root of the problem size.

scholarly journals Complexity of n-Queens Completion (Extended Abstract)

2018 ◽  
Author(s):  
Ian P. Gent ◽  
Christopher Jefferson ◽  
Peter Nightingale
Keyword(s):  
Artificial Intelligence ◽  
Phase Transition ◽  
Decision Problem ◽  
Completion Problem ◽  
Complete Problems ◽  
Random Instances ◽  
Np Complete

The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

Statistical properties of hybrid composites. I. Recursion analysis

1983 ◽  
Vol 389 (1796) ◽  
pp. 67-100 ◽  
Keyword(s):  
Hybrid Composites ◽  
Probability Model ◽  
Load Sharing ◽  
Upper Bounds ◽  
Strength Distribution ◽  
Local Load ◽  
Sharing Rule ◽  
Critical Crack ◽  
Breaking Strain ◽  
Hybrid Effect

An adaptation of the chain-of-bundles probability model for unidirectional intraply hybrid composites consisting of two types of fibres is given. Local load sharing, which is sensitive to the different elastic moduli of the fibres, is assumed for the non-failed fibre segments in each bundle. A sequence of tight upper bounds is developed for the probability distribu­tion of strength for the hybrid. The upper bounds are based upon the occurrence of k or more adjacent broken fibre segments in a bundle; this event is necessary but not sufficient for bundle failure. This development allows for a description of a critical crack size k *, dependent upon the load on the hybrid, which is a characterization of the length of a crack that catastrophically propagates causing bundle failure with virtual certainty. The upper bound developed with k *, based upon the hybrid median strength, is essentially identical to the true probability distribution of hybrid strength. It is also shown that the strength distribution for the hybrid composite has a weakest link structure in terms of a charac­teristic distribution function that is highly dependent upon the local load sharing rule, the fibre properties, and the geometrical structure of the hybrid. Numerical results from the model show that typically there is a negative ‘hybrid effect’ for hybrid breaking strain, but there is a positive ‘hybrid effect’ for hybrid tensile strength.

scholarly journals Improving WalkSAT for Random 3-SAT Problems

2020 ◽  
Vol 26 (2) ◽  
pp. 220-243
Author(s):  
Huimin Fu ◽  
Yang Xu ◽  
Shuwei Chen ◽  
Jun Liu
Keyword(s):  
Phase Transition ◽  
Ant Colony Algorithm ◽  
State Of The Art ◽  
Stochastic Local Search ◽  
Improved Genetic Algorithm ◽  
Initial Assignment ◽  
Sat Solver ◽  
Random Instances ◽  
Improved Ant Colony Algorithm ◽  
Better Than

Stochastic local search (SLS) algorithms are well known for their ability to efficiently find models of random instances of the Boolean satisfiability (SAT) problems. One of the most famous SLS algorithms for SAT is called WalkSAT, which has wide influence and performs well on most of random 3-SAT instances. However, the performance of WalkSAT lags far behind on random 3-SAT instances equal to or greater than the phase transition ratio. Motivated by this limitation, in the present work, firstly an allocation strategy is introduced and utilized in WalkSAT to determine the initial assignment, leading to a new algorithm called WalkSATvav. The experimental results show that WalkSATvav significantly outperforms the state-of-the-art SLS solvers on random 3-SAT instances at the phase transition for SAT Competition 2017. However, WalkSATvav cannot rival its competitors on random 3-SAT instances greater than the phase transition ratio. Accordingly, WalkSATvav is further improved for such instances by utilizing a combination of an improved genetic algorithm and an improved ant colony algorithm, which complement each other in guiding the search direction. The resulting algorithm, called WalkSATga, is far better than WalkSAT and significantly outperforms some previous known SLS solvers on random 3-SAT instances greater than the phase transition ratio from SAT Competition 2017. Finally, a new SAT solver called WalkSATlg, which combines WalkSATvav and WalkSATga, is proposed, which is competitive with the winner of random satisfiable category of SAT competition 2017 on random 3-SAT problem.

scholarly journals A phase transition in the distribution of the length of integer partitions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Dimbinaina Ralaivaosaona
Keyword(s):  
Phase Transition ◽  
Positive Integer ◽  
Gaussian Distribution ◽  
Gumbel Distribution ◽  
Limiting Distribution ◽  
Integer Partitions ◽  
Random Partition ◽  
Uniform Probability ◽  
International Audience

International audience We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.

The Satisfiability Threshold Conjecture: Techniques Behind Upper Bound Improvements

2005 ◽  
Author(s):  
Lefteris M. Kirousis ◽  
Lefteris M. Stamatiou
Keyword(s):  
Phase Transition ◽  
Lower Bounds ◽  
Fixed Ratio ◽  
Theoretical Work ◽  
Upper And Lower Bounds ◽  
Crossover Point ◽  
Previous Part ◽  
Boolean Formulas ◽  
Lower Ratio ◽  
Almost All

One of the most challenging problems in probability and complexity theory is to establish and determine the satisfiability threshold, or phase transition, for random 3-SAT instances: Boolean formulas consisting of clauses with exactly k literals. As the previous part of the volume has explored, empirical observations suggest that there exists a critical ratio of the number of clauses to the number of variables, such that almost all randomly generated formulas with a higher ratio are unsatisfiable while almost all randomly generated formulas with a lower ratio are satisfiable. The statement that such a crossover point really exists is called the satisfiability threshold conjecture. Experiments hint at such a direction, but as far as theoretical work is concerned, progress has been difficult. In an important advance, Friedgut [177] showed that the phase transition is a sharp one, though without proving that it takes place at a “fixed” ratio for large formulas. Otherwise, rigorous proofs have focused on providing successively better upper and lower bounds for the value of the (conjectured) threshold. In this chapter, our goal is to review the series of improvements of upper bounds for 3-SAT and the techniques leading to these. We give only a passing reference to the improvements of the lower bounds as they rely on significantly different techniques, one of which is discussed in the next chapter. Let ϕ be a random k-SAT formula constructed by selecting, uniformly and with replacement, ra clauses from the set of all possible clauses with k literals (no variable repetitions allowed within a clause) over n variables. It has been experimentally observed that as the numbers m, n of variables and clauses tend to infinity while the ratio or clause density m/n is fixed to a constant a, the property of satisfiability exhibits a phase transition. For the case of 3-SAT, when a is greater than a number that has been experimentally determined to be approximately α < 4.27, then almost all random 3-SAT formulas are unsatisfiable; that is, the fraction of unsatisfiable formulas tends to 1.

scholarly journals Covering n-Permutations with (n+1)-Permutations

10.37236/2168 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Taylor F Allison ◽  
Anant P Godbole ◽  
Kathryn M Hawley ◽  
Bill Kay
Keyword(s):  
Phase Transition ◽  
Upper Bounds ◽  
Probability Of Coverage

Let $S_n$ be the set of all permutations on $[n]:=\{1,2,\ldots,n\}$. We denote by $\kappa_n$ the smallest cardinality of a subset ${\cal A}$ of $S_{n+1}$ that "covers" $S_n$, in the sense that each $\pi\in S_n$ may be found as an order-isomorphic subsequence of some $\pi'$ in ${\cal A}$.  What are general upper bounds on $\kappa_n$?  If we randomly select $\nu_n$ elements of $S_{n+1}$, when does the probability that they cover $S_n$ transition from 0 to 1?  Can we provide a fine-magnification analysis that provides the "probability of coverage"  when $\nu_n$ is around the level given by the phase transition?   In this paper we answer these questions and raise others.

scholarly journals Measure of unevenness in human genomes, described as a self-affine phase transition in a 'spin-chain’ model

F1000Research ◽  
2021 ◽  
Vol 10 ◽  
pp. 149
Author(s):  
Sergey Feranchuk
Keyword(s):  
Phase Transition ◽  
Gaussian Distribution ◽  
Probability Model ◽  
Chain Model ◽  
Scientific Investigations ◽  
Human Genomes ◽  
A Genome ◽  
Spin Chain Model ◽  
History Of ◽  
Non Gaussian

Background: Non-Gaussian distribution of polymorphic positions across a genome can substantially influence the results of any approach to molecular evolution based on a 'classical' probability model. The infinite dispersion of non-Gaussian perturbations is a challenge in an attempt to accept it in a probability-based model of evolution. Methods: Here a model is proposed where non-Gaussian distribution is introduced to an exact solution of the 'Ising model'; it describes a behavior of one-dimensional chain of spins in an approaching to a phase transition. The distribution of fragments which are identical between two genomes is similar to distribution of islands of spins with the same orientation, in the model where non-integer dimension is introduced. Results: Application of this model allows us to compare the relative contributions of non-Gaussian perturbations for pairs of human genomes from different ethnic groups. An evolution of the three human races in a most compact presentation is considered, rates of development on the separated stages of the evolution are assumed to be proportional to a value of relative unevenness between the appropriate groups of genomes. In the resolved model, the meaningful details of the separation between Asian and European races are clarified, in a period around ten thousand years ago; a particular viewpoint to the separation of the African race is also presented. Conclusion: The proposed approximation of non-Gaussian perturbations in human genomes allows to support the statements which are otherwise missed in the scientific investigations of the early history of modern humans.

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