Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Holant Problems

  • Jin-Yi CaiEmail author
  • Heng Guo
  • Tyson Williams
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_748

Years and Authors of Summarized Original Work

  • 2011; Cai, Lu, Xia

  • 2012; Huang, Lu

  • 2013; Cai, Guo, Williams

  • 2013; Guo, Lu, Valiant

  • 2014; Cai, Guo, Williams

Problem Definition

The framework of Holant problems is intended to capture a class of sum-of-product computations in a more refined way than counting CSP problems and is inspired by Valiant’s holographic algorithms [12] (also cf. entry  Holographic Algorithms). A constraint function f, or signature, is a mapping from [κ]n to \(\mathbb{C}\)


Computational complexity Counting complexity Holant problems Partition functions 
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Recommended Reading

  1. 1.
    Cai JY, Lu P (2011) Holographic algorithms: from art to science. J Comput Syst Sci 77(1):41–61MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cai JY, Lu P, Xia M (2010) Holographic algorithms with matchgates capture precisely tractable planar #CSP. In: FOCS, Las Vegas. IEEE Computer Society, pp 427–436Google Scholar
  3. 3.
    Cai JY, Lu P, Xia M (2011) Computational complexity of Holant problems. SIAM J Comput 40(4):1101–1132MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cai JY, Lu P, Xia M (2011) Dichotomy for Holant* problems of Boolean domain. In: SODA, San Francisco. SIAM, pp 1714–1728Google Scholar
  5. 5.
    Cai JY, Huang S, Lu P (2012) From Holant to #CSP and back: dichotomy for Holantc problems. Algorithmica 64(3):511–533MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cai JY, Guo H, Williams T (2013) A complete dichotomy rises from the capture of vanishing signatures (extended abstract). In: STOC, Palo Alto. ACM, pp 635–644zbMATHGoogle Scholar
  7. 7.
    Cai JY, Lu P, Xia M (2013) Dichotomy for Holant* problems with domain size 3. In: SODA, New Orleans. SIAM, pp 1278–1295Google Scholar
  8. 8.
    Cai JY, Guo H, Williams T (2014) The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems. In: FOCS, Philadelphia. IEEE, pp 601–610Google Scholar
  9. 9.
    Guo H, Lu P, Valiant LG (2013) The complexity of symmetric Boolean parity Holant problems. SIAM J Comput 42(1):324–356MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Huang S, Lu P (2012) A dichotomy for real weighted Holant problems. In: CCC, Porto. IEEE Computer Society, pp 96–106Google Scholar
  11. 11.
    Valiant LG (2006) Accidental algorithms. In: FOCS, Berkeley. IEEE, pp 509–517Google Scholar
  12. 12.
    Valiant LG (2008) Holographic algorithms. SIAM J Comput 37(5):1565–1594MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Beijing UniversityBeijingChina
  2. 2.Computer Sciences Department, University of Wisconsin–MadisonMadisonUSA
  3. 3.Computer Sciences Department, University of Wisconsin–MadisonMadisonUSA