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Efficient computation of rank probabilities in posets

Karel De Loof (UGent)
(2009)
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Abstract
As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced. A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present. The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven. The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed. The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be.
Keywords
posets, linear extensions, lattice of ideals, rank probabilities, weak order extensions, mutual rank probabilities, cycle transitivity, proportional transitivity, minimum cutting levels, monotone data sets, algorithms

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Citation

Please use this url to cite or link to this publication:

Chicago
De Loof, Karel. 2009. “Efficient Computation of Rank Probabilities in Posets”. Gent.
APA
De Loof, K. (2009). Efficient computation of rank probabilities in posets. Gent.
Vancouver
1.
De Loof K. Efficient computation of rank probabilities in posets. [Gent]; 2009.
MLA
De Loof, Karel. “Efficient Computation of Rank Probabilities in Posets.” 2009 : n. pag. Print.
@phdthesis{874495,
  abstract     = {As the title of this work indicates, the central theme in this work is the computation of rank probabilities of posets. Since the probability space consists of the set of all linear extensions of a given poset equipped with the uniform probability measure, in first instance we develop algorithms to explore this probability space efficiently. We consider in particular the problem of counting the number of linear extensions and the ability to generate extensions uniformly at random. Algorithms based on the lattice of ideals representation of a poset are developed. 

Since a weak order extension of a poset can be regarded as an order on the equivalence classes of a partition of the given poset not contradicting the underlying order, and thus as a generalization of the concept of a linear extension, algorithms are developed to count and generate weak order extensions uniformly at random as well. However, in order to reduce the inherent complexity of the problem, the cardinalities of the equivalence classes is fixed a priori. Due to the exponential nature of these algorithms this approach is still not always feasible, forcing one to resort to approximative algorithms if this is the case. It is well known that Markov chain Monte Carlo methods can be used to generate linear extensions uniformly at random, but no such approaches have been used to generate weak order extensions. Therefore, an algorithm that can be used to sample weak order extensions uniformly at random is introduced.

A monotone assignment of labels to objects from a poset corresponds to the choice of a weak order extension of the poset. Since the random monotone assignment of such labels is a step in the generation process of random monotone data sets, the ability to generate random weak order extensions clearly is of great importance. The contributions from this part therefore prove useful in e.g. the field of supervised classification, where a need for synthetic random monotone data sets is present.

The second part focuses on the ranking of the elements of a partially ordered set. Algorithms for the computation of the (mutual) rank probabilities that avoid having to enumerate all linear extensions are suggested and applied to a real-world data set containing pollution data of several regions in Baden-Württemberg (Germany). With the emergence of several initiatives aimed at protecting the environment like the REACH (Registration, Evaluation, Authorisation and Restriction of Chemicals) project of the European Union, the need for objective methods to rank chemicals, regions, etc. on the basis of several criteria still increases. Additionally, an interesting relation between the mutual rank probabilities and the average rank probabilities is proven.

The third and last part studies the transitivity properties of the mutual rank probabilities and the closely related linear extension majority cycles or LEM cycles for short. The type of transitivity is translated into the cycle-transitivity framework, which has been tailor-made for characterizing transitivity of reciprocal relations, and is proven to be situated between strong stochastic transitivity and a new type of transitivity called delta*-transitivity. It is shown that the latter type is situated between strong stochastic transitivity and a kind of product transitivity. Furthermore, theoretical upper bounds for the minimum cutting level to avoid LEM cycles are found. Cutting levels for posets on up to 13 elements are obtained experimentally and a theoretic lower bound for the cutting level to avoid LEM cycles of length 4 is computed.

The research presented in this work has been published in international peer-reviewed journals and has been presented on international conferences. A Java implementation of several of the algorithms presented in this work, as well as binary files containing all posets on up to 13 elements with LEM cycles, can be downloaded from the website http://www.kermit.ugent.be.},
  author       = {De Loof, Karel},
  keywords     = {posets,linear extensions,lattice of ideals,rank probabilities,weak order extensions,mutual rank probabilities,cycle transitivity,proportional transitivity,minimum cutting levels,monotone data sets,algorithms},
  language     = {eng},
  pages        = {174},
  school       = {Ghent University},
  title        = {Efficient computation of rank probabilities in posets},
  year         = {2009},
}