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From Graph Colouring to Constraint Satisfaction: There and Back Again

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Book cover Topics in Discrete Mathematics

Part of the book series: Algorithms and Combinatorics ((AC,volume 26))

Abstract

Graph colourings may be viewed as special constraint satisfaction problems. The class of k-colouring problems enjoys a well known dichotomy of complexity — these problems are polynomial time solvable when k ≤ 2, and NP-complete when k ≥ 3. For general constraint satisfaction problems such dichotomy was conjectured by Feder and Vardi, but has still not been proved in full generality. We discuss some results and techniques related to this Dichotomy Conjecture. We focus on the effects of a new concept of ‘fullness’, and how it affects the complexity of constraint satisfaction problems and their dichotomy. Full constraint satisfaction problems may then be specialized back to graph colourings, yielding an interesting novel class of problems in graph theory, related to the study of graph perfection.

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References

  1. B. Aspvall, F. Plass and R. E. Tarjan, A linear time algorithm for testing the truth of certain quantified Boolean formulas, Information Processing Letters, Vol. 8 (1979), 121–123.

    Article  MATH  MathSciNet  Google Scholar 

  2. H.-J. Bandelt, A. Dählmann, H. Schütte, Absolute retracts of bipartite graphs, Discrete Appl. Math. 16 (1987), 191–215.

    Article  MATH  MathSciNet  Google Scholar 

  3. H.-J. Bandelt, M. Farber, and P. Hell, Absolute reflexive retracts and absolute bipartite retracts, Discrete Applied Math. 44 (1993), 9–20.

    Article  MATH  MathSciNet  Google Scholar 

  4. H.-J. Bandelt, E. Pesch, Efficient characterizations of n-chromatic absolute retracts, J. Combinatorial Theory B 53 (1991), 5–31.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Bang-Jensen and P. Hell, The effect of two cyles on the complexity of colourings by directed graphs, Discrete Applied Math. 26 (1990), 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Bang-Jensen, P. Hell, and G. MacGillivray, The complexity of colouring by semicomplete digraphs, SIAM J. on Discrete Math. 1 (1988), 281–298.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Bang-Jensen, P. Hell, and G. MacGillivray, On the complexity of colouring by superdigraphs of bipartite graphs, Discrete Math. 109 (1992), 27–44.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Bang-Jensen, P. Hell, and G. MacGillivray, Hereditarily hard H-colouring problems, Discrete Math. 138 (1995), 75–92.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Birc, M. Hujter, and Zs. Tuza, Precoloring extension I; interval graphs, Discrete Math. 100 (1992), 267–279.

    Article  MathSciNet  Google Scholar 

  10. R. Brewster, T. Feder, P. Hell, J. Huang, and G. MacGillivray, Near unanimity functions and varieties of graphs, manuscript 2004.

    Google Scholar 

  11. K. Borsuk, Sur les rétractes, Fund. Math. 17 (1931), 152–170.

    MATH  Google Scholar 

  12. A. Branstädt, P. Hammer, V. Bang Le, and V. Lozin, Bisplit graphs, DIM ACS Technical Report 2002-44 (2002).

    Google Scholar 

  13. A. A. Bulatov, Tractable conservative constraint satisfaction problems, In Proceedings of the 18th IEEE Annual Symposium on Logic in Computer Science (LICS 2003), 321–330.

    Google Scholar 

  14. A. A. Bulatov, Maltsev constraints are tractable, Technical report PRG RR-02-05 Oxford University (2002).

    Google Scholar 

  15. A. A. Bulatov, A dichotomy constraint on a three-element set, In Proceedings of the 43rd IEEE Symposium on Theory of Computing (2002), 649–658.

    Google Scholar 

  16. A. A. Bulatov, H-colouring dichotomy revisited, Theoretical Computer Science, to appear.

    Google Scholar 

  17. A. A. Bulatov and P. Jeavons, Algebraic structures in combinatorial problems, Technical Report MATH-AL-4-2001m Technische univer-sität Dresden, Germany, 2001.

    Google Scholar 

  18. A. A. Bulatov, P. Jeavons, and A. A. Krokhin, Classifying complexity of constraints using finite algebras, SIAM J. on Computing 34 (2005), 720–742.

    Article  MATH  MathSciNet  Google Scholar 

  19. K. Cameron, E. E. Eschen, C. T. Hoang and R. Sritharan, The list partition problem for graphs, Proc. 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2004, 391–399.

    Google Scholar 

  20. E. Cheng, R. P. Kleinberg, S. G. Kruk, W. A. Lindsey, and D. E. Steffy, A strictly combinatorial approach to a university exam scheduling problem, Congr. Numer. 167 (2004) 121–132.

    MATH  MathSciNet  Google Scholar 

  21. M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, to appear in the Annals of Math.

    Google Scholar 

  22. V. Chvátal, Star-cutsets and perfect graphs, J. Combinatorial Theory B 39 (1985), 189–199.

    Article  MATH  Google Scholar 

  23. D. Cohen, M. Cooper, and P. Jeavons, Characterizing tractable constraints, Artificial Intelligence 65 (1994), 347–361.

    Article  MATH  MathSciNet  Google Scholar 

  24. N. Creignou, S. Khanna, and M. Sudan, Complexity Classifications of Boolean Constraint Satisfaction Problems, SIAM Monographs on Discrete Math. and Applications, vol. 7 (2001).

    Google Scholar 

  25. V. Dalmau, A new tractable class of constraint satisfaction problems, in Proceedings 6th International Symposium on Artificial Intelligence and Mathematics, 2000.

    Google Scholar 

  26. V. Dalmau and D. Ford, Generalized satisfiability with k occurrences per variable: a study through delta-matroid parity, Mathematical Foundations of Computer Science (MFCS 2003), Lecture Notes in Computer Science 2747, Springer 2003, 358–367.

    MathSciNet  Google Scholar 

  27. R. Dechter, Constraint networks, In Encyclopedia of Artificial Intelligence, 1992, 276–285.

    Google Scholar 

  28. W. F. Dowling and J. H. Gallier, Linear-time algorithms for testing the satisfiability of propositional Horn formulae, J. Logic Programming 1 (1984) 267–284.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Feder, Fanout limitations on constraint systems, Theoretical Computer Science 255 (2001), 281–293.

    Article  MATH  MathSciNet  Google Scholar 

  30. T. Feder, Homomorphisms to oriented cycles and k-partite satisfiability, SIAM J. Discrete Math. 14 (2001), 471–480.

    Article  MATH  MathSciNet  Google Scholar 

  31. T. Feder, Constraint satisfaction: a personal perspective, manuscript 2004.

    Google Scholar 

  32. T. Feder, A dichotomy theorem on fixed points of several nonexpansive mappings, to appear in SIAM J. Discrete Math.

    Google Scholar 

  33. T. Feder and D. Ford, Classification of bipartite Boolean constraint satisfaction through delta matroid intersection, manuscript.

    Google Scholar 

  34. T. Feder and P. Hell, List homomorphisms to reflexive graphs, J. Comb. Theory Series B 72 (1998), 236–250.

    Article  MathSciNet  Google Scholar 

  35. T. Feder and P. Hell, Full constraint satisfaction problems, to appear in SIAM J. on Computing.

    Google Scholar 

  36. T. Feder and P. Hell, Matrix partitions of perfect graphs, to appear in Berge Special Issue of Discrete Math.

    Google Scholar 

  37. T. Feder and P. Hell, On realizations of point determining graphs, and obstructions to full homomorphisms, manuscript 2004.

    Google Scholar 

  38. T. Feder and P. Hell, The complexity of retraction problems for reflexive digraphs, manuscript 2003.

    Google Scholar 

  39. T. Feder, P. Hell, and J. Huang, List homomorphisms and circular arc graphs, Combinatorica 19 (1999), 487–505.

    Article  MATH  MathSciNet  Google Scholar 

  40. T. Feder, P. Hell, and J. Huang, Bi-arc graphs and the complexity of list homomorphisms, J. Graph Theory 42 (1999), 61–80.

    Article  MathSciNet  Google Scholar 

  41. T. Feder, P. Hell, and J. Huang, List homomorphisms of graphs with bounded degrees, to appear in Discrete Math.

    Google Scholar 

  42. T. Feder, P. Hell, and J. Huang, List homomorphisms to reflexive digraphs, manuscript 2004.

    Google Scholar 

  43. T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of list partitions, SIAM J. Discrete Mathematics 16 (2003), 449–478.

    Article  MATH  MathSciNet  Google Scholar 

  44. T. Feder, P. Hell, S. Klein, L. Nogueira, and F. Protti, List matrix partitions of chordal graphs, Theoretical Computer Science 349 (2005), 52–66.

    MATH  MathSciNet  Google Scholar 

  45. T. Feder, P. Hell, D. Král’, and J. Sgall, Two algorithms for list matrix partitions, Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2005, 870–876.

    Google Scholar 

  46. T. Feder, P. Hell, and B. Mohar, Acyclic homomorphisms and circular colorings of digraphs, SIAM J. Discrete Math. 17 (2003), 161–169.

    Article  MATH  MathSciNet  Google Scholar 

  47. T. Feder, P. Hell, and K. Tucker-Nally, Digraph matrix partitions and trigraph homomorphisms, manuscript 2004.

    Google Scholar 

  48. T. Feder, P. Hell, and W. Hochstättler, Generalized colourings (matrix partitions) of cographs, GT04 Special Berge Volume, to be published by Birkhauser.

    Google Scholar 

  49. T. Feder, F. Madelaine, and I. A. Stewart, Dichotomies for classes of homomorphism problems involving unary functions, Theoretical Computer Science 314 (2004), 1–43.

    Article  MATH  MathSciNet  Google Scholar 

  50. T. Feder and M. Y. Vardi, The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory, SI AM J. Gomput. 28 (1998), 236–250, (ALSO in STOC 25 (1993), 612–622).

    Google Scholar 

  51. C. M. H. de Figueiredo, S. Klein, Y. Kohayakawa and B. Reed, Finding skew partitions efficiently, J. Algorithms 37 (2000), 505–521.

    Article  MATH  MathSciNet  Google Scholar 

  52. A. Galluccio, P. Hell, J. Nešetřil, The complexity of H-colouring of bounded degree graphs, Discrete Math. 222 (2000), 101–109.

    Article  MATH  MathSciNet  Google Scholar 

  53. M. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-completeness, Freeman and Company, 1979.

    Google Scholar 

  54. M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.

    MATH  Google Scholar 

  55. W. Gutjahr, E. Welzl and G. Woeginger, Polynomial graph-colorings, Discrete Applied Math 35 (1992), 29–45.

    Article  MATH  MathSciNet  Google Scholar 

  56. P. Hell, Rétractions des graphes, PhD thesis, Université de Montreal, 1972.

    Google Scholar 

  57. P. Hell, Absolute planar retracts and the four color conjecture, J. Combinatorial Theory B 17 (1974), 5–10.

    Article  MATH  MathSciNet  Google Scholar 

  58. P. Hell, Absolute retracts in graphs, in Graphs and Combinatorics (R. A. Bari, F. Harary, eds.), Springer-Verlag Lecture Notes in Mathematics 406 (1974), 291–301.

    Google Scholar 

  59. P. Hell, S. Klein, L. Tito Nogueira, and F. Protti, Partitioning chordal graphs into independent sets and cliques, Discrete Applied Math. 141 (2004), 185–194.

    Article  MATH  Google Scholar 

  60. P. Hell, S. Klein, L. Tito Nogueira, and F. Protti, Packing r-cliques in chordal graphs, Annals of O.R. 138 (2005), 179–187.

    Article  MATH  Google Scholar 

  61. P. Hell, J. Nešetřil, The core of a graph, Discrete Math. 109 (1992), 117–126.

    Article  MATH  MathSciNet  Google Scholar 

  62. P. Hell and J. Nešetřil, On the complexity of H-coloring, J. Comb. Theory, Series B 48 (1990), 92–110.

    Article  MATH  Google Scholar 

  63. P. Hell and J. Nešetřil, Counting list homomorphisms and graphs with bounded degrees, in Graphs, Morphisms and Statistical Physics (J. Nešetřil and P. Winkler, eds.) DIM ACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 63 (2004), 105–112.

    Google Scholar 

  64. P. Hell and J. Nešetřil, Graphs and Homomorphisms, Oxford University Press 2004.

    Google Scholar 

  65. P. Hell, J. Nešetřil, and X. Zhu, Complexity of tree homomorphisms, Discrete Applied Math. 70 (1996), 23–36.

    Article  MATH  Google Scholar 

  66. P. Hell, I. Rival, Absolute retracts and varieties of reflexive graphs, Canad. J. Math. 39 (1987), 544–567.

    MATH  MathSciNet  Google Scholar 

  67. A. Horn, On sentences which are true of direct unions of algebras, J. Symbolic Logic 16 (1951), 14–21.

    Article  MATH  MathSciNet  Google Scholar 

  68. M. Hujter and Zs. Tuza, Precoloring extension II; graph classes related to bipartite graphs, Acta Math. Universitatis Comenianae 62 (1993), 1–11.

    MATH  MathSciNet  Google Scholar 

  69. M. Hujter and Zs. Tuza, Precoloring extension III; classes of perfect graphs, Combin. Probab. Comput. 5 (1996), 35–56.

    Article  MATH  MathSciNet  Google Scholar 

  70. E. M. Jawhari, D. Misane, M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, Contemp. Math. (Amer. Math. Soc.) 57 (1986), 175–226.

    MathSciNet  Google Scholar 

  71. P. Jeavons, On the algebraic structure of combinatorial problems, Theoretical Computer Science 200 (1998), 185–204.

    Article  MATH  MathSciNet  Google Scholar 

  72. P. Jeavons, D. Cohen, and M. Gyssens, Closure properties of constraints, J. of the ACM 44 (1997), 527–548.

    Article  MATH  MathSciNet  Google Scholar 

  73. P. Jeavons, D. Cohen, and M. Gyssens, A unifying framework for tractable constraints, Proc. 1st International Conference on Constraint Programming, CP′95, Cassis 1995.

    Google Scholar 

  74. D. S. Johnson, The NP-completeness columns: an ongoing guide, J. Algorithms 3 (1982), 89–99.

    Article  MATH  MathSciNet  Google Scholar 

  75. V. Kumar, Algorithms for constraint-satisfaction problems, AI Magazine 13 (1992), 32–44.

    Google Scholar 

  76. R. E. Ladner, On the structure of polynomial time reducibility, J. Assoc. Comput. Mach. 22 (1975), 155–171.

    MATH  MathSciNet  Google Scholar 

  77. B. Larose, Taylor operations on finite reflexive structures, manuscript 2005.

    Google Scholar 

  78. B. Larose and C. Tardif, Projectivity and independent sets in powers of graphs, J. Graph Theory 40 (2002), 162–171.

    Article  MATH  MathSciNet  Google Scholar 

  79. B. Larose and L. Zádori, Finite posets and topological spaces in locally finite varieties, Algebra Universalis 52 (2004), 119–136.

    Article  MATH  MathSciNet  Google Scholar 

  80. B. Larose and L. Zádori, The complexity of the extendibility problem for finite posets, SIAM J. on Discrete Math. 17 (2003), 114–121.

    Article  MATH  Google Scholar 

  81. P. Lincoln and J. C. Mitchell, Algorithmic aspects of type inference with subtypes, in Conf. Rec. 19th ACM Symp. on Principles of Programming Languages (1992), 293–304.

    Google Scholar 

  82. V. Lotfi and S. Sarin, A graph coloring algorithm for large scale scheduling problems, Comput. Oper. Res. 13 (1986), 27–32.

    Article  MATH  MathSciNet  Google Scholar 

  83. T. Łuczak, J, Nešetřil, A note on projective graphs, ITI series 2003, No. 130.

    Google Scholar 

  84. T. Łuczak, J. Nešetřil, Towards probabilistic analysis of the dichotomy problem, KAM-DIMATIA Series 2003–640, Charles Univ. Prague.

    Google Scholar 

  85. D. Marx, Precoloring extension on chordal graphs, manuscript 2003.

    Google Scholar 

  86. P. Meseguer, Constraint satisfaction problem: an overview, AICOM 2 (1989), 3–16.

    Google Scholar 

  87. J. C. Mitchell, Coercion and type inference (summary), in Conf. Rec. 11th ACM Symp. on Principles of Programming Languages (1984), 175–185.

    Google Scholar 

  88. U. Montanari, Networks of constraints: Fundamental properties and applications to picture processing, Information Sciences 7 (1974), 95–132.

    Article  MathSciNet  Google Scholar 

  89. E. Pesch, Retracts of Graphs, Athenaeum Verlag, Frankfurt 1988.

    MATH  Google Scholar 

  90. V. Pratt and J. Tiuryn, Satisfiability of inequalities in a poset, Fundamenta Informaticae 28 (1996), 165–182.

    MATH  MathSciNet  Google Scholar 

  91. A. Pultr, V. Trnková, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  92. A. Quilliot, A retraction problem in graph theory, Discrete Math. 54 (1985), 61–71.

    Article  MATH  MathSciNet  Google Scholar 

  93. A. Salazar and R. V. Oakford, A graph formulation of a school scheduling algorithm, Comm. ACM 17 (1974), 696–698.

    Article  MATH  MathSciNet  Google Scholar 

  94. T. J. Schaefer, The complexity of satisfiability problems, Proc. 10th ACM Symp. on Theory of Computing (1978), 216–226.

    Google Scholar 

  95. R. E. Tarjan, Decomposition by clique separators, Discrete Mathematics 55 (1985), 221–232.

    Article  MATH  MathSciNet  Google Scholar 

  96. E. P. K. Tsang, Foundations of Constraint Satisfaction, Academic Press, London and San Diego (1993).

    Google Scholar 

  97. M. Y. Vardi, Constraint satisfaction and database theory: a tutorial, Proceedings of the 19th Symposium on Principles of Database Systems (PODS 2000), 76–85.

    Google Scholar 

  98. N. Vikas, Computational complexity of compaction to reflexive cycles SIAM J. on Computing 32 (2003), 253–280.

    Article  MATH  MathSciNet  Google Scholar 

  99. M. Wand and P. M. O’Keefe, On the complexity of type inference with coercion, in Conf. on Functional Programming Languages and Computer Architecture (1989), 293–298.

    Google Scholar 

  100. W. Xie, Forbidden subgraph characterizations of matrix partitions, M.Sc. thesis, Simon Eraser University 2006.

    Google Scholar 

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  1. School of Computing Science, Simon Eraser University, Burnaby, B.C., V5A 1S6, Canada

    Pavol Hell

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  1. Pavol Hell
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  1. Department of Applied Mathematics and Institute for Theoretical Computer Science ITI, Faculty of Mathematics and Physics, Charles University, Prague, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Martin Klazar , Jan Kratochvíl , Martin Loebl , Jiří Matoušek  & Pavel Valtr , , ,  & 

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Robin Thomas

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Hell, P. (2006). From Graph Colouring to Constraint Satisfaction: There and Back Again. In: Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Valtr, P., Thomas, R. (eds) Topics in Discrete Mathematics. Algorithms and Combinatorics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-33700-8_20

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