Journal of Classification

, Volume 34, Issue 2, pp 191–222 | Cite as

Robinsonian Matrices: Recognition Challenges



Ultrametric inequality is involved in different operations on (dis)similarity matrices. Its coupling with a compatible ordering leads to nice interpretations in seriation problems. We accurately review the interval graph recognition problem for its tight connection with recognizing a dense Robinsonian dissimilarity (precisely, in the anti-ultrametric case). Since real life matrices are prone to errors or missing entries, we address the sparse case and make progress towards recognizing sparse Robinsonian dissimilarities with lexicographic breadth first search. The ultrametric inequality is considered from the same graph point of view and the intimate connection between cocomparability graph and dense Robinsonian similarity is established. The current trend in recognizing special graph structures is examined in regard to multiple lexicographic search sweeps. Teaching examples illustrate the issues addressed for both dense and sparse symmetric matrices.


Robinsonian matrices Interval graphs Cocomparability graphs Lexicographic searches Partition refinement 


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  1. AVIS, D., and FUKUDA, K. (1992), “A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra”, Discrete and Computational Geometry, 8(3), 295–313.MathSciNetCrossRefMATHGoogle Scholar
  2. BERRY, A., and BORDAT, J.-P. (1998), “Separability Generalizes Dirac’s Theorem”, Discrete and Applied Mathematics, 84(1-3), 43–53.MathSciNetCrossRefMATHGoogle Scholar
  3. BOOTH, K.S., and LUEKER, G.S. (1976), “Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms”, Journal of Computer and System Sciences, 13(3), 335–379.MathSciNetCrossRefMATHGoogle Scholar
  4. BURKARD, R.E., ÇELA, E., ROTE, G., and WOEGINGER, G.J. (1998), “The Quadratic Assignment Problem with a Monotone Anti-Monge and a Symmetric Toeplitz Matrix: Easy and Hard Cases”, Mathematical Programming, 82(1-2, Ser. B), 125–158.MathSciNetCrossRefMATHGoogle Scholar
  5. ÇELA, E., DEINEKO, V.G., and WOEGINGER, G.J. (2012), “Another Well-Solvable Case of the QAP: Maximizing the Job Completion Time Variance”, Operations Research Letters, 40(5), 356–359.MathSciNetCrossRefMATHGoogle Scholar
  6. CHEPOI, V., and FICHET, B. (1997), “Recognition of Robinsonian Dissimilarities”, Journal of Classification, 14(2), 311–325.MathSciNetCrossRefMATHGoogle Scholar
  7. CHEPOI, V., FICHET, B., and SESTON, M. (2009), “Seriation in the Presence of Errors: NP-Hardness of l∞-Fitting Robinson Structures to Dissimilarity Matrices”, Journal of Classification, 26(3), 279–296.MathSciNetCrossRefMATHGoogle Scholar
  8. CORNEIL, D.G. (2004), “A Simple 3-Sweep LBFS Algorithm for the Recognition of Unit Interval Graphs”, Discrete Applied Mathematics, 138(3), 371–379.MathSciNetCrossRefMATHGoogle Scholar
  9. CORNEIL, D.G., OLARIU, S., and STEWART, L. (1999), “Linear Time Algorithms for Dominating Pairs in Asteroidal Triple-Free Graphs”, SIAM Journal on Computing, 28(4), 1284–1297.MathSciNetCrossRefMATHGoogle Scholar
  10. CORNEIL, D.G., OLARIU, S. and STEWART, L., (2009/10) “The LBFS Structure and Recognition of Interval Graphs”, SIAM Journal on Discrete Mathematics, 23(4), 1905–1953.MathSciNetCrossRefMATHGoogle Scholar
  11. CORNEIL, D.G., DUSART, J., HABIB, M., MAMCARZ, A., and DE MONTGOLFIER, F. (2016), “A Tie-Break Model for Graph Search”, Discrete Applied Mathematics, 199, 89–100.MathSciNetCrossRefMATHGoogle Scholar
  12. DAHLHAUS, E. (1993), “Fast Parallel Recognition of Ultrametrics and TreeMetrics”, SIAM Journal on Discrete Mathematics, 6(4), 523–532.MathSciNetCrossRefMATHGoogle Scholar
  13. DIETRICH, B.L. (1990), “Monge Sequences, Antimatroids, and the Transportation Problem with Forbidden Arcs”, Linear Algebra and Its Applications, 139, 133–145.MathSciNetCrossRefMATHGoogle Scholar
  14. FORTIN, D., and RUDOLF, R. (1998), “Weak Monge Arrays in Higher Dimensions”, Discrete Mathematics, 189(1-3), 105–115.MathSciNetCrossRefMATHGoogle Scholar
  15. HABIB, M., MCCONNELL, R., PAUL, C., and VIENNOT, L. (2000), “Lex-BFS and Partition Refinement, with Applications to Transitive Orientation, Interval Graph Recognition and Consecutive Ones Testing”, Theoretical Computer Science, 234(1-2), 59–84.MathSciNetCrossRefMATHGoogle Scholar
  16. HOŞTEN, S., and MORRIS, W.D. Jr. (1999), “The Order Dimension of the Complete Graph”, Discrete Mathematics, 201(1-3), 133–139.MathSciNetCrossRefMATHGoogle Scholar
  17. HUBERT, L., and ARABIE, P. (1994), “The Analysis of Proximity Matrices Through Sums of Matrices Having (Anti)-Robinson Forms”, British Journal of Mathematical and Statistical Psychology, 47, 1–40.CrossRefMATHGoogle Scholar
  18. KLINZ, B., RUDOLF, R., and WOEGINGER, G.J. (1995), “On the Recognition of Permuted Bottleneck Monge Matrices”, Discrete Applied Mathematics, 63(1), 43–74.MathSciNetCrossRefMATHGoogle Scholar
  19. KÖHLER, E., and MOUATADID, L. (2014), “Linear Time LexDFS on Cocomparability Graphs”, in Algorithm Theory—SWAT 2014, Vol. 8503, Lecture Notes in Computer Science, Springer, pp. 319–330.Google Scholar
  20. LAURENT, M. (1996), “Graphic Vertices of the Metric Polytope”, Discrete Mathematics, 151(1-3), 131–153.MathSciNetCrossRefMATHGoogle Scholar
  21. LAURENT, M., and SEMINAROTI, M. (2015a), A Lex-BFS-Based Recognition Algorithm for Robinsonian Matrices, Springer International Publishing, pp. 325–338.Google Scholar
  22. LAURENT, M., and SEMINAROTI, M. (2015b), “The Quadratic Assignment Problem is Easy for Robinsonian Matrices with Toeplitz Structure”, Operations Research Letters, 43(1), 103–109.MathSciNetCrossRefGoogle Scholar
  23. LAURENT, M., and SEMINAROTI, M. (2016), “Similarity-First Search: A New Algorithm with Application To RobinsonianMatrix Recognition”, CoRR, abs/1601.03521,
  24. LEKKERKERKER, C.G., and BOLAND, J.C. (1962/1963), “Representation of a Finite Graph by a Set of Intervals on the Real Line”, Fundamenta Mathematicae, 51, 45–64.MathSciNetMATHGoogle Scholar
  25. LI, P., and WU, Y. (2014), “A Four-Sweep LBFS Recognition Algorithm for Interval Graphs”, Discrete Mathematics and Theoretical Computer Science, 16(3), 23–50.MathSciNetMATHGoogle Scholar
  26. MCCONNELL, R.M., and SPINRAD, J.P. (1994), “Linear-Time Modular Decomposition and Efficient Transitive Orientation of Comparability Graphs”, in Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), New York: ACM, pp. 536–545.Google Scholar
  27. PRÉA, P., and FORTIN, D. (2014), “An Optimal Algorithm to Recognize Robinsonian Dissimilarities”, Journal of Classification, 31(3), 351–385.MathSciNetCrossRefMATHGoogle Scholar
  28. ROBINSON, W. (1951), “A Method for Chronologically Ordering Archaeological Deposits”, American Antiquity, 16(4), 293–301.CrossRefGoogle Scholar
  29. RUDOLF, R., and WOEGINGER, G.J. (1995), “The Cone of Monge Matrices: Extremal Rays and Applications”, ZOR—Mathematical Methods of Operations Research, 42(2), 161–168.MathSciNetCrossRefMATHGoogle Scholar
  30. TSEVENDORJ, I. (2001), “Piecewise-Convex Maximization Problems: Global Optimality Conditions”, Journal of Global Optimization, 21(1), 1–14.MathSciNetCrossRefMATHGoogle Scholar
  31. XU, S.-J., LI, X., and LIANG, R. (2013), “Moplex Orderings Generated by the LexDFS Algorithm”, Discrete Applied Mathematics, 161(13-14), 2189–2195.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Classification Society of North America 2017

Authors and Affiliations

  1. 1.INRIALe Chesnay cedexFrance

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