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Complexity of Constraints pp 255–282Cite as

Introduction to the Maximum Solution Problem

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5250))

Abstract

This paper surveys complexity and approximability results for the Maximum Solution (Max Sol) problem. Max Sol is an optimisation variant of the constraint satisfaction problem. Many important and well-known combinatorial optimisation problems are instances of Max Sol: for example, Max Sol restricted to the domain {0,1} is exactly the Max Ones problem (which, in turn, captures problems such as Independent Set and 0/1 Integer Programming). By using this relationship, many different problems in logic, graph theory, integer programming, and algebra can be given a uniform treatment. This opens up for new ways of analysing and solving combinatorial optimisation problems.

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References

  1. Ageev, A.: On finding critical independent and vertex sets. SIAM J. Discrete Math. 7(2), 293–295 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti Spaccamela, A., Protasi, M.: Complexity and approximation: Combinatorial optimization problems and their approximability properties. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  3. Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41, 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodirsky, M., Nešetřil, J.: Constraint satisfaction with countable homogeneous templates. Journal of Logic and Computation 16(3), 359–373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bulatov, A.: A graph of a relational structure and constraint satisfaction problems. In: Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004), pp. 448–457 (2004)

    Google Scholar 

  6. Bulatov, A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM 53(1), 66–120 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34(3), 720–742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bulatov, A., Krokhin, A., Jeavons, P.: The complexity of maximal constraint languages. In: Proceedings of the 33rd ACM Symposium on Theory of Computing (STOC 2001), pp. 667–674 (2001)

    Google Scholar 

  9. Cohen, D., Cooper, M., Jeavons, P.: An algebraic characterisation of complexity for valued constraint. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 107–121. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  10. Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: The complexity of soft constraint satisfaction. Artificial Intelligence 170(11), 983–1016 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Creignou, N., Hermann, M., Krokhin, A., Salzer, G.: Complexity of clausal constraints over chains. Theory Comput. Syst. 42(2), 239–255 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of Boolean constraint satisfaction problems. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  13. Crescenzi, P., Silvestri, R., Trevisan, L.: On weighted vs unweighted versions of combinatorial optimization problems. Inf. Comput. 167(1), 10–26 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dalmau, V.: A new tractable class of constraint satisfaction problems. Annals of Mathematics and Artificial Intelligence 44(1-2), 61–85 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gil, À., Hermann, M., Salzer, G., Zanuttini, B.: Efficient algorithms for constraint description problems over finite totally ordered domains. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS, vol. 3097, pp. 244–258. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  18. Goldmann, M., Russell, A.: The complexity of solving equations over finite groups. In: IEEE Conference on Computational Complexity, pp. 80–86 (1999)

    Google Scholar 

  19. Gutin, G., Hell, P., Rafiey, A., Yeo, A.: A dichotomy for minimum cost graph homomorphisms. European J. Combin. 29(4), 900–911 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gutin, G., Rafiey, A., Yeo, A.: Minimum cost and list homomorphisms to semicomplete digraphs. Discrete Applied Mathematics 154(6), 890–897 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gutin, G., Rafiey, A., Yeo, A., Tso, M.: Level of repair analysis and minimum cost homomorphisms of graphs. Discrete Applied Mathematics 154(6), 881–889 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hell, P., Nešetřil, J.: On the complexity of H-colouring. Journal of Combinatorial Theory B 48, 92–110 (1990)

    Article  MATH  Google Scholar 

  23. Hähnle, R.: Complexity of many-valued logics. In: Proceedings of the 31st IEEE International Symposium on Multiple-valued Logic (ISMVL 2001), pp. 137–148 (2001)

    Google Scholar 

  24. Hochbaum, D., Megiddo, N., Naor, J., Tamir, A.: Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Mathematical Programming 62, 69–84 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hochbaum, D., Naor, J.: Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput. 23(6), 1179–1192 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. Journal of the ACM 48(4), 761–777 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. Journal of the ACM 44, 527–548 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jeavons, P., Cooper, M.: Tractable constraints on ordered domains. Artificial Intelligence 79, 327–339 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jonsson, P.: Boolean constraint satisfaction: complexity results for optimization problems with arbitrary weights. Theoretical Computer Science 244(1-2), 189–203 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jonsson, P., Klasson, M., Krokhin, A.: The approximability of three-valued Max CSP. SIAM J. Comput. 35(3), 1329–1349 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Jonsson, P., Kuivinen, F., Nordh, G.: Max Ones generalised to larger domains. SIAM J. Comput. 38(1), 329–365 (2008)

    Article  MATH  Google Scholar 

  33. Jonsson, P., Nordh, G.: Generalised integer programming based on logically defined relations. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 549–560. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  34. Jonsson, P., Nordh, G., Thapper, J.: The maximum solution problem on graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 228–239. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  35. Khanna, S., Sudan, M., Trevisan, L., Williamson, D.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kuivinen, F.: Tight approximability results for the maximum solution equation problem over Z p . In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 628–639. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  37. Pöschel, R., Kaluznin, L.: Funktionen- und Relationenalgebren. DVW, Berlin (1979)

    Google Scholar 

  38. Rosenberg, I.: Minimal clones I: the five types. In: Szabó, L., Szendrei, Á. (eds.) Lectures in Universal Algebra. North-Holland, Amsterdam (1986)

    Google Scholar 

  39. Schrijver, A.: A combinatorial algorithm for minimizing submodular functions in polynomial time. Journal of Combinatorial Theory B 80, 346–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Szczepara, B.: Minimal clones generated by groupoids. PhD thesis, Université de Montréal (1996)

    Google Scholar 

  41. Szendrei, Á.: Clones in Universal Algebra. In: Séminaires de Mathématiques Supérieures, University of Montreal, vol. 99 (1986)

    Google Scholar 

  42. Woeginger, G.: An efficient algorithm for a class of constraint satisfaction problems. Operations Research Letters 30(1), 9–16 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, C.: Finding critical independent sets and critical vertex subsets are polynomial problems. SIAM J. Discrete Math. 3(3), 431–438 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proceedings of the 38th ACM Symposium on Theory of Computing (STOC 2006), pp. 681–690 (2006)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer and Information Science, Linköpings Universitet, S-581 83, Linköpings, Sweden

    Peter Jonsson

  2. Laboratoire d’Informatique, École Polytechnique, F-91128, Palaiseau, France

    Gustav Nordh

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  1. Peter Jonsson
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  2. Gustav Nordh
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Editor information

Editors and Affiliations

  1. LIF (CNRS UMR 6166), Aix-Marseille Université, 163 avenue de Luminy, F-13288, Marseille, France

    Nadia Creignou

  2. IBM Almaden Research Center, Computer Science Principles and Methodologies, and Computer Science Department, University of California, Santa Cruz, USA

    Phokion G. Kolaitis

  3. Institut für Theoretische Informatik, Leibniz, Universität Hannover, Appelstr. 4, 30167, Hannover, Germany

    Heribert Vollmer

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Jonsson, P., Nordh, G. (2008). Introduction to the Maximum Solution Problem. In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds) Complexity of Constraints. Lecture Notes in Computer Science, vol 5250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92800-3_10

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  • DOI: https://doi.org/10.1007/978-3-540-92800-3_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-92799-0

  • Online ISBN: 978-3-540-92800-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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