A branch-and-bound algorithm for fitting anti-robinson structures to symmetric dissimilarity matrices

Brusco, Michael J.

doi:10.1007/bf02294996

Cited by 15 publications

(30 citation statements)

References 21 publications

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“…In particular, since C C is not a (G2)-arc, we must have d z 4 d y . Analogously, since C C is not a (G3)-arc, we deduce that d(x, y) 16 d (x, z).…”

Section: Lemma 417mentioning

confidence: 97%

“…As an error measure one can use the usual l p -distance between two matrices of equal size. Various heuristics for approximate seriation using Robinson matrices have been considered in [16,39,40] and papers cited therein (the package seriation [36] contains an implementation of several such methods). In particular, Hubert [39] noticed: "Whether or not certain seriation techniques produce 'better' approximation to a Robinson matrix is essentially an open question and requires .…”

Section: Seriation Problemmentioning

confidence: 99%

“…Further suppose that d(x, x ) 2 d(x, z). Since d(x , z) 16 d (x, z), in any -compatible order the cell C must be located between the cells C and C. Now, additionally assume that d(y , z) 2 d (x, z). Then d(y , z) 14 d (x , z).…”

Section: Lemma 413mentioning

confidence: 99%

“…Lemma 4.17 yields d (x, y) 16 d (x, z). Suppose by way of contradiction that d(y, z) 16 d (x, z). If yz ∈ L ij , then d(y, z) 3 max{d y , d z }, i.e.…”

Section: Lemma 418mentioning

confidence: 99%

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Seriation in the Presence of Errors: A Factor 16 Approximation Algorithm for l ∞-Fitting Robinson Structures to Distances

Chepoi

Seston

2009

Algorithmica

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The classical seriation problem consists in finding a permutation of the rows and the columns of the distance (or, more generally, dissimilarity) matrix d on a finite set X so that small values should be concentrated around the main diagonal as close as possible, whereas large values should fall as far from it as possible. This goal is best achieved by considering the Robinson property: a distance d R on X is Robinsonian if its matrix can be symmetrically permuted so that its elements do not decrease when moving away from the main diagonal along any row or column. If the distance d fails to satisfy the Robinson property, then we are lead to the problem of finding a reordering of d which is as close as possible to a Robinsonian distance.In this paper, we present a factor 16 approximation algorithm for the following NP-hard fitting problem: given a finite set X and a dissimilarity d on X, we wish to find a Robinsonian dissimilarity d R on X minimizing the l ∞ -error d − d R ∞ = max x,y∈X {|d(x, y) − d R (x, y)|} between d and d R .

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“…In particular, since C C is not a (G2)-arc, we must have d z 4 d y . Analogously, since C C is not a (G3)-arc, we deduce that d(x, y) 16 d (x, z).…”

Section: Lemma 417mentioning

confidence: 97%

Section: Seriation Problemmentioning

confidence: 99%

Section: Lemma 413mentioning

confidence: 99%

“…Lemma 4.17 yields d (x, y) 16 d (x, z). Suppose by way of contradiction that d(y, z) 16 d (x, z). If yz ∈ L ij , then d(y, z) 3 max{d y , d z }, i.e.…”

Section: Lemma 418mentioning

confidence: 99%

See 2 more Smart Citations

Seriation in the Presence of Errors: A Factor 16 Approximation Algorithm for l ∞-Fitting Robinson Structures to Distances

Chepoi

Seston

2009

Algorithmica

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“…Consequently, the desired permutations in the collection, 1 , , K ρ ρ * *  , are also at least locally-optimal. Alternative strategies for solving the QA problem include dynamic programming [10] and branch-and-bound algorithms [11] [12].…”

Section: Optimal Ar Decomposition: Algorithmic and Computational Detailsmentioning

confidence: 99%

Anti-Robinson Structures for Analyzing Three-Way Two-Mode Proximity Data

Köhn¹

2014

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Multiple discrete (non-spatial) and continuous (spatial) structures can be fitted to a proximity matrix to increase the information extracted about the relations among the row and column objects vis-à-vis a representation featuring only a single structure. However, using multiple discrete and continuous structures often leads to ambiguous results that make it difficult to determine the most faithful representation of the proximity matrix in question. We propose to resolve this dilemma by using a nonmetric analogue of spectral matrix decomposition, namely, the decomposition of the proximity matrix into a sum of equally-sized matrices, restricted only to display an order-constrained patterning, the anti-Robinson (AR) form. Each AR matrix captures a unique amount of the total variability of the original data. As our ultimate goal, we seek to extract a small number of matrices in AR form such that their sum allows for a parsimonious, but faithful reconstruction of the total variability among the original proximity entries. Subsequently, the AR matrices are treated as separate proximity matrices. Their specific patterning lends them immediately to the representation by a single (discrete non-spatial) ultrametric cluster dendrogram and a single (continuous spatial) unidimensional scale. Because both models refer to the same data base and involve the same number of parameters, estimated through least-squares, a direct comparison of their differential fit is legitimate. Thus, one can readily determine whether the amount of variability associated which each AR matrix is most faithfully represented by a discrete or a continuous structure, and which model provides in sum the most appropriate representation of the original proximity matrix. We propose an extension of the order-constrained anti-Robinson decomposition of square-symmetric proximity matrices to the analysis of individual differences of three-way data, with the third way representing individual data sources. An application to judgments of schematic face stimuli illustrates the method.

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Minimizing Sign Changes Rowwise: Consecutive Ones Property and Beyond

Fortin

Tseveendorj²

2015

Analysis, Modelling, Optimization, and Numerical Techniques

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A branch-and-bound algorithm for fitting anti-robinson structures to symmetric dissimilarity matrices

Abstract: combinatorial optimization, branch and bound, seriation, anti-Robinson form,

Cited by 15 publications

References 21 publications

Seriation in the Presence of Errors: A Factor 16 Approximation Algorithm for l ∞-Fitting Robinson Structures to Distances

Seriation in the Presence of Errors: A Factor 16 Approximation Algorithm for l ∞-Fitting Robinson Structures to Distances

Anti-Robinson Structures for Analyzing Three-Way Two-Mode Proximity Data

Minimizing Sign Changes Rowwise: Consecutive Ones Property and Beyond

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