Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Complexity Dichotomies for Counting Graph Homomorphisms

  • Jin-Yi CaiEmail author
  • Xi Chen
  • Pinyan Lu
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_747

Years and Authors of Summarized Original Work

  • 2000; Dyer, Greenhill

  • 2005; Bulatov, Grohe

  • 2010; Goldberg, Grohe, Jerrum, Thurley

  • 2013; Cai, Chen, Lu

Problem Definition

It is well known that if NP ≠ P, there is an infinite hierarchy of complexity classes between them [10]. However, for some broad classes of problems, a complexity dichotomy exists: every problem in the class is either in polynomial time or NP-hard. Such results include Schaefer’s theorem [13], the dichotomy of Hell and Nešetřil for H-coloring [9], and some subclasses of the general constraint satisfaction problem [4]. These developments lead to the following questions: How far can we push the envelope and show dichotomies for even broader classes of problems? Given a class of problems, what is the criterion that distinguishes the tractable problems from the intractable ones? How does it help in solving the tractable problems efficiently? Now replacing NP with #P [15], all the questions above can be asked for counting...


Computational complexity Counting complexity Graph homomorphisms Partition functions 
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Bulatov A, Grohe M (2005) The complexity of partition functions. Theor Comput Sci 348(2):148–186MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cai JY, Chen X, Lu P (2013) Graph homomorphisms with complex values: a dichotomy theorem. SIAM J Comput 42(3):924–1029MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Carlitz L (1969) Kloosterman sums and finite field extensions. Acta Arith 16:179–193MathSciNetzbMATHGoogle Scholar
  4. 4.
    Creignou N, Khanna S, Sudan M (2001) Complexity classifications of boolean constraint satisfaction problems. SIAM monographs on discrete mathematics and applications. Society for Industrial and Applied Mathematics, PhiladelphiazbMATHCrossRefGoogle Scholar
  5. 5.
    Dyer M, Greenhill C (2000) The complexity of counting graph homomorphisms. Random Struct Algorithms 17(3–4):260–289MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Freedman M, Lovász L, Schrijver A (2007) Reflection positivity, rank connectivity, and homomorphism of graphs. J Am Math Soc 20:37–51MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Goldberg L, Grohe M, Jerrum M, Thurley M (2010) A complexity dichotomy for partition functions with mixed signs. SIAM J Comput 39(7):3336–3402MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Grohe M, Thurley M (2011) Counting homomorphisms and partition functions. In: Grohe M, Makowsky J (eds) Model theoretic methods in finite combinatorics. Contemporary mathematics, vol 558. American Mathematical Society, ProvidenceGoogle Scholar
  9. 9.
    Hell P, Nešetřil J (1990) On the complexity of H-coloring. J Comb Theory Ser B 48(1):92–110MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ladner R (1975) On the structure of polynomial time reducibility. J ACM 22(1):155–171MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lidl R, Niederreiter H (1997) Finite fields. Encyclopedia of mathematics and its applications, vol 20. Cambridge University Press, CambridgeGoogle Scholar
  12. 12.
    Linial N (1986) Hard enumeration problems in geometry and combinatorics. SIAM J Algebraic Discret Methods 7:331–335MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Schaefer T (1978) The complexity of satisfiability problems. In: Proceedings of the 10th annual ACM symposium on theory of computing, San Diego, California, pp 216–226Google Scholar
  14. 14.
    Vadhan S (2002) The complexity of counting in sparse, regular, and planar graphs. SIAM J Comput 31:398–427MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Valiant L (1979) The complexity of computing the permanent. Theor Comput Sci 8:189–201MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Valiant L (1979) The complexity of enumeration and reliability problems. SIAM J Comput 8:410–421MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Beijing UniversityBeijingChina
  2. 2.Computer Sciences DepartmentUniversity of Wisconsin–MadisonMadisonUSA
  3. 3.Computer Science DepartmentColumbia UniversityNew YorkUSA
  4. 4.Computer Science and TechnologyTsinghua UniversityBeijingChina
  5. 5.Microsoft Research AsiaShanghaiChina