Achlioptas, D., L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou: 1997, `Random Constraint Satisfaction: a More Accurate Picture'. In: Proceedings CP'97. Linz, Austria, pp. 107--120.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Cerf et al. New results Table 1. Upper time bound for various algorithms for binary csps of the phase transition. Also note that the very existence of phase transitions has been questioned, cf. [1]. Thus the complexity given in Cerf et al. cannot be directly compared with ours. The structure of the paper is as follows: Section 2 contains some basic information on constraint satisfaction and quantum computation needed for the algorithm, which is presented and described in Section 3. Here we ....

D. Achlioptas, L. M. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: A more accurate picture. Constraints, 6(4):329--344, 2001.

....sizes, but there is not yet any proof that the transition happens for the same parameter value for different sizes. Note that some of the methods commonly used to generate more complicated random constraint satisfaction problems have been shown not to have a transition in the thermodynamic limit [81]. The transition can be seen in figure 3.1, 33 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 connectivity g fraction of solved problems Figure 3.1: This figure illustrates the phase transition in solvability that occurs at # #:# for the # COL ....

D. Achlioptas, L. M. Kirousis, E. Kranakis, D. Krizanc, M. S. O. Molloy and Y. C. Stamatiou, Random Constraint Satisfaction: A More Accurate Picture,inProceedings of Third International Conference on Principles and Practice of Constraint Programming, vol. 1330, pp. 107--120. Springer, Berlin, 1997.

....wrongly instantiated variables) different re nement functions, and decay functions, so as to give a complete overview of di erent variations on this algorithm. Another novelty in the present work is the usage of a recently proposed problem instance generator. As shown by Achlioptas et al. in [1], the widely applied CSP generators (much used in EA research) have serious de ciencies. Most importantly, the generated instances tend to be asymptotically unsolvable, preventing a sound study of algorithmic behavior around the phase transition. The proposed alternative cures this problem and ....

....have arity k = 2 (bound two variables) and where all the variable domains contain the same number of values D. We adhere to this simpli cation because it restricts experimental complexity and every CSP of arity larger than two has an equivalent binary CSP ( 17] 3 GENERATING CSPS In [1], Achlioptas et al. show that the so called Models A to D are unsuitable for the study of phase transition and threshold phenomena such as CSPs. This is because the instances they asymptotically generate have almost certainly no solutions. A general framework for these models, presented in [15, ....

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D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M.S.O. Molloy, and Y.C. Stamatiou. Random constraint satisfaction: A more accurate picture. In G. Smolka, editor, Principles and Practice of Constraint Programming | CP97, number 1330 in Lecture Notes in Computer Science, pages 107-120, Berlin, 1997. SpringerVerlag.

....have arity k = 2 (bound two variables) and where all the variable domains contain the same number of values D. We adhere to this simpli cation because it restricts experimental complexity and every CSP of arity larger than two has an equivalent binary CSP ( 6] III. CSP GENERATORS In [1], Achlioptas et al. show that the so called Models A to D (de ned below) are unsuitable for the study of phase transition and threshold phenomena such as CSPs. This is because the instances they asymptotically generate have almost certainly no solutions. A general framework for these models, ....

....the options for the two sets, we get four slightly di erent models for generating random CSPs, in particular, in the terminology used in [5] if both Step 1 and 2 are done with option (i) we get Model A, while if both steps are done with option (ii) we get Model B. As Achlioptas et al. show in [1] Model A generates almost certainly unsatis able instances for every p 2 6= 0, while Model B generates almost certainly unsatis able instances for every p 2 1=D (analogously for the other two models) In the same paper an alternative model for generating random CSP instances is proposed. The ....

D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M.S.O. Molloy, and Y.C. Stamatiou. Random constraint satisfaction: A more accurate picture. In G. Smolka, editor, Principles and Practice of Constraint Programming | CP97, number 1330 in Lecture Notes in Computer Science, pages 107-120, Berlin, 1997. Springer-Verlag.

....occurs, then this model will be asymptotically uninteresting for study. Since the phase transition phenomena were found in k SAT and some other combinatorial problems [4] random CSP has also received great attention in recent years, both from an experimental and a theoretical point of view [1], 7] 8] 12] 13] 16] 17] 18] However, there is still some lack of studies about the probabilistic analysis of random CSP models. This paper mainly analyzes the average complexity of backtracking on random constraint satisfaction problems. In section 1, we first give a brief introduction of ....

D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M. S.O. Molloy, Y. C. Stamatiou. Random Constraint Satisfaction: A More Accurate Picture. Submitted to Constraints. Also In Proc. Third International Conference on Principles and Practice of Constraint Programming (CP-97), SpringerVerlag, ages 107--120, 1997.

....numerous applications ranging from vision, language comprehension to scheduling and diagnosis [7] In general, CSP is NP complete. Recently, there has been a great amount of interest in the phase transition behaviour of random CSPs, both from an experimental and a theoretical point of view [2,8,18,20,21,23,24] . However, there is still some lack of studies about the structure of solutions of random CSPs. To study the phase transition behaviour of random CSPs, we need first a random CSP model to generate random instances. Standard Model B [8,20] the most commonly used CSP model in previous studies, ....

....tuples of values. The parameter r determines how many constraints are in a CSP instance, while q determines how restrictive the constraints are. Assume that in Model GB all the variable domains contain the same number of values 2 d and q satisfies d q . Recently, Achlioptas et al. [2] shows that for standard Model B, i.e. the binary case of Model GB, if d q , almost all the instances generated following Model B are trivially insoluble as the number of variables approaches infinity. Following the same lines as their proof for Model B we can easily show that if d q , ....

Achlioptas, D., Kirousis, L. M., Kranakis, E., Krizanc, D., Molloy, M. S.O., & Stamatiou, Y. C. Random constraint satisfaction: a more accurate picture. Constraints, to appear. Also in Proceedings of CP97, pages 107-120, Springer, 1997.

....threshold criterion for monotone subsets of the hypercube. In using this criterion he proved that k SAT exhibits a sharp threshold for k 2 but without specifying its location. In the last few years there has been a great amount of interest in constraint satisfaction problems (see [SD96] and [AKKKMS98]) In [AKKKMS98] the authors proposed a model for generating random constraint satisfaction instances which exhibit a non trivial asymptotic behavior. They also suggested to study models where there is some structure, or even no randomness at all, in the choice of forbidden pairs of values for two ....

....for monotone subsets of the hypercube. In using this criterion he proved that k SAT exhibits a sharp threshold for k 2 but without specifying its location. In the last few years there has been a great amount of interest in constraint satisfaction problems (see [SD96] and [AKKKMS98] In [AKKKMS98] the authors proposed a model for generating random constraint satisfaction instances which exhibit a non trivial asymptotic behavior. They also suggested to study models where there is some structure, or even no randomness at all, in the choice of forbidden pairs of values for two constrained ....

D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M. Molloy and Y.C. Stamatiou. Random constraint satisfaction: a more accurate picture. Random structures and algorithms, 12:253--269, 1998.

....sample of random instances for each class. For each sample, the median or average solving cost is computed. However, our generic constraint store features a non binary constraint, so we cannot literally apply this characterisation of instance classes. In any case, the latter has been criticised [1] because it is unrealistic to have a constant tightness p 2 for all constraints, so that many possible instances can never be generated. For these two reasons, we developed the following characterisation of instance classes, which is speci c to the considered family. It is not subject to any of ....

....unrealistic to have a constant tightness p 2 for all constraints, so that many possible instances can never be generated. For these two reasons, we developed the following characterisation of instance classes, which is speci c to the considered family. It is not subject to any of the criticisms in [1], because it exploits the structure of the generic constraint store. The generic nite domain constraint store for the SubsetB family is parameterised by the number n of Boolean variables involved (i.e. the size of the given set T ) and the given size k of the sought subset S, and contains an ....

D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M.S.O. Molloy, and Y.C. Stamatiou. Random constraint satisfaction: A more accurate picture. In: G. Smolka (ed), Proc. of CP'97, pp. 107-120. LNCS 1330. Springer-Verlag, 1997.

....on the accuracy and for which problem instances does this hold. 4 Research in progress To better test the techniques that we develop we use a di erent model for randomly generating binary constraint satisfaction problems. This model, called model E, behaves much better when we scale up problems [1]. We are applying this new model in a large scale comparison of every evolutionary technique that solves constraint satisfaction problems we have seen over the past ten years. With the same model we are testing an extended version of the Stepwise Adaptation of Weights technique that we have named ....

D. Achlioptas, L.M. Kirousis, E.K., D. Krizanc, M. S.O. Molloy, and Y.C. Stamatiou. Random constraint satisfaction a more accurate picture. In Gert Smolka, editor, Principles and Practice of Constraint Programming | CP97, pages 107{

....the probability of getting the universal relation is very low, we ignore this in the following. 295 Renz Nebel This way of generating random instances is very similar to the way random CSP instances over nite domains are usually generated (Gent, MacIntyre, Prosser, Smith, Walsh, 2001) Achlioptas et al. 1997) found that the standard models for generating random CSP instances over nite domains lead to trivially awed instances for n 1, i.e. instances become locally inconsistent without having to propagate constraints. Since we are using CSP instances over in nite domains, Achlioptas et al. s result ....

....tractable subsets. A theoretical analysis of the two models showed that the model H and the model A for a small average degree of the nodes in the constraint graph do not su er from trivial local inconsistencies as it is the case for similar generation procedures for CSPs with nite domains (Achlioptas et al. 1997). It turned out that randomly generated instances of both models show a phase transition behavior which depends most strongly on the average degree of the instances. While most instances outside the phase transition region can be 313 Renz Nebel solved eciently by each of our heuristics, ....

Achlioptas, D., Kirousis, L., Kranakis, E., Krizanc, D., Molloy, M., & Stamatiou, Y. (1997). Random constraint satisfaction: a more accurate picture. In 3rd Conference on the Principles and Practice of Constraint Programming (CP'97), Vol. 1330 of LNCS, pp. 107-120. Springer-Verlag.

.... of the phase transition in random 3 Sat narrows as problems increase in size [15] However, we only have rather loose but hard won bounds on its actual location [16, 37] For random constraint satisfaction problems (CSPs) Achlioptas et al. recently provided a more negative theoretical result [1]. They show that, as the number of variables increases, the conventional random models produce problems which almost surely contain awed variables and are therefore trivially insoluble. Thus, these models do not have an asymptotic phase transition over most of their parameter space. This paper ....

....While we use the same notation p 1 and p 2 in each model, note that in some cases the value is used as a proportion, and in others as a probability. For example in model D, p 1 is used as a proportion but p 2 is used as a probability. 3 The Problem with Random Problems Achlioptas et al. [1] identify a shortcoming of all four random models. They prove that if p 2 1=m then, as n 1, there almost surely exists a awed variable, one for which every value is awed. A value for a variable is awed if, when the value is assigned to the variable, there exists an adjacent variable in the ....

[Article contains additional citation context not shown here]

D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M.S.O. Molloy, and Y.C. Stamatiou. Random constraint satisfaction: A more accurate picture. In Proc. CP97, pages 107-120. Springer, 1997.

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D. Achlioptas, L. M. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: a more accurate picture. Constraints. To appear.

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D. Achlioptas, L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: a more accurate picture. Constraints 6, 329 - 324 (2001). Conference version in Proceedings of CP 97, 107 - 120.

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Dimitris Achlioptas, Lefteris M. Kirousis, Evangelos Kranakis, Danny Krizanc, Michael Molloy, and Yiannis Stamatiou, Random constraint satisfaction: a more accurate picture, 3rd Conference on the Principles and Practice of Constraint Programming (Linz, Austria,

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D. Achlioptas, L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: a more accurate picture. Constraints 6, 329 - 324 (2001). Conference version in Proceedings of CP 97, 107 - 120.

....is kept constant as the number of variables increases. The very active experimental study of random models of CSP has necessitated a rigorous analysis of such models. Various models of random CSP s for which m, the domain size, is constant have been studied in several papers, for example [2, 21, 11, 22, 23]. One of the earliest such studies, 2] discovered that the most natural models su er a fatal aw (described below) The rst study of the case where m grows with n was [13] where one of these most natural models was studied. Implicit in that study was the fact that for certain settings of the ....

....The very active experimental study of random models of CSP has necessitated a rigorous analysis of such models. Various models of random CSP s for which m, the domain size, is constant have been studied in several papers, for example [2, 21, 11, 22, 23] One of the earliest such studies, [2] discovered that the most natural models su er a fatal aw (described below) The rst study of the case where m grows with n was [13] where one of these most natural models was studied. Implicit in that study was the fact that for certain settings of the relevant parameters, the fatal aw did ....

[Article contains additional citation context not shown here]

D. Achlioptas, L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: a more accurate picture. Constraints 6, 329 - 324 (2001). Conference version in Proceedings of CP 97, 107 - 120.

....with probability p = c Thetak n k Gamma1 . This variation produces models which are equivalent to Models A1 and A2 in the sense that the models will be a.s. satisfiable and a.s. unsatisfiable for the same values of c. A similar remark applies to the other two models described below. In [5], we observed that Model A1 is asymptotically uninteresting in the sense that as long as M grows with n, the random CSP will be a.s. unsatisfiable. The problem is quite simple: a.s. there will be at least one constraint which is overconstraining in that there is no assignment to its variables ....

D. Achlioptas, L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: a more accurate picture. (submitted) .

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Achlioptas, D., L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou: 1997, `Random Constraint Satisfaction: a More Accurate Picture'. In: Proceedings CP'97. Linz, Austria, pp. 107--120.

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D. Achlioptas, L. M. Kirousis, E. Kranakis, D. Krizanc, M. S. O. Molloy, and Y. C. Stamatiou. Random constraint satisfaction: A more accurate picture. In Principles and Practice of Constraint Programming, pages 107--120, 1997.

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Achlioptas, D., Kirousis, L., Kranakis, E., Krizanc, D., Molloy, M., Stamatiou, Y.: Random Constraint Satisfaction: a More Accurate Picture. In: Proceedings CP'97, Linz, Austria (1997) 107--120

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D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M.S.O. Molloy, and Y.C. Stamatiou, `Random constraint satisfaction: A more accurate picture', in Proc. CP97, pp. 107--120. Springer, (1997).

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Achlioptas, D., Kirousis, L., Kranakis, E., Krizanc, D., Molloy, M., Stamatiou, Y.: Random constraint satisfaction: A more accurate picture. Constraints 4 (2001) 329--344

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Achlioptas, Kirousis, Kranakis, Krizanc, Molloy, and Stamatiou. Random constraint satisfaction: A more accurate picture. In ICCP: International Conference on Constraint Programming (CP), LNCS, 1997.

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D. Achlioptas, L. Kirousis, E. Kranakis, D. Krizanc, M. Molloy, and Y. Stamatiou. Random constraint satisfaction: a more accurate picture. In Proceedings of CP'97, pages 107--120, 1997.

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D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, M.S.O. Molloy, and C. Stamatiou. Random constraint satisfaction: A more accurate picture. In Proceedings of Third International Conference on Principles and Practice of Constraint Programming (CP97), pages 107--120, 1997.

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