Sub-dominant theory in numerical taxonomy

Brucker, François

doi:10.1016/j.dam.2005.09.011

Cited by 8 publications

(12 citation statements)

References 7 publications

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“…Figure 10 shows a diagonal quatuor (a), a lateral quatuor (b) and a quatuor which is both lateral and diagonal (c). Moreover, the following result by Brucker (2002) completes the characterization of the relationships between quasiultrametric quatuors and 2-ultrametric quatuors.…”

Section: Quasi-ultrametric Quatuorsmentioning

confidence: 86%

“…Assertions (ii) and (iii) are due to Brucker (2001Brucker ( , 2002. Durand (1989) had previously noticed that strongly Robinsonian dissimilarities do not admit subdominants (when restricted to strongly Robinsonian dissimilarities with a given compatible order, they nevertheless admit weak subdominants).…”

Section: A Subdominant Theoremmentioning

confidence: 90%

“…More generally, it can be shown that the set of all strongly Robinsonian dissimilarities admits lower maximal dissimilarities (for the proof see Brucker 2001Brucker , 2002. We omit the corresponding construction.…”

Section: The Case Of Strongly Robinsonian Dissimilaritiesmentioning

confidence: 97%

“…Note that a quatuor can be both diagonal and lateral and that lateral or diagonal quatuors are not supposed to be 2-ultrametric (all the edge weights may differ). However it can be shown (Brucker 2002) that a diagonal quatuor is quasiultrametric if and only if it is 2-ultrametric. Figure 10 shows a diagonal quatuor (a), a lateral quatuor (b) and a quatuor which is both lateral and diagonal (c).…”

Section: Quasi-ultrametric Quatuorsmentioning

confidence: 97%

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Combinatorial optimisation and hierarchical classifications

2007

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This paper is devoted to some selected topics relating Combinatorial Optimization and Hierarchical Classification. It is oriented toward extensions of the standard classification schemes (the hierarchies): pyramids, quasi-hierarchies, circular clustering, rigid clustering and others. Bijection theorems between these models and dissimilarity models allow to state some clustering problems as optimization problems. Within the galaxy of optimization we have especially discussed the following: NP-completeness results and search for polynomial instances; problems solved in a polynomial time (e.g. subdominant theory); design, analysis and applications of algorithms. In contrast with the orientation to "new" clustering problems, the last part discusses some standard algorithmic approaches.

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Section: Quasi-ultrametric Quatuorsmentioning

confidence: 86%

Section: A Subdominant Theoremmentioning

confidence: 90%

Section: The Case Of Strongly Robinsonian Dissimilaritiesmentioning

confidence: 97%

Section: Quasi-ultrametric Quatuorsmentioning

confidence: 97%

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Combinatorial optimisation and hierarchical classifications

2007

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“…In particular this clearly holds for A =R (p,q) with 1 ≤ p ≤ q ≤ n. Notice that, in general, the permuted matrix A π * is not an anti-Monge matrix any more. Figure 6 illustrates the effect of permuting the 10 × 10 matrixR (2,7) according to permutation π * .…”

Section: The Up-benevolent Qapmentioning

confidence: 99%

New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

Çela

Deı̆neko

Woeginger

2018

European Journal of Operational Research

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We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.

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Weak Hierarchies: A Central Clustering Structure

Bertrand

Diatta

2014

Clusters, Orders, and Trees: Methods and Applications

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Sub-dominant theory in numerical taxonomy

Cited by 8 publications

References 7 publications

Combinatorial optimisation and hierarchical classifications

Combinatorial optimisation and hierarchical classifications

New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

Weak Hierarchies: A Central Clustering Structure

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