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...

Keywords

Computational complexity Counting complexity Graph homomorphisms Partition functions 
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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