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Encyclopedia of Algorithms pp 921–926Cite as

Holographic Algorithms

  • Reference work entry
  • First Online:

  • 101 Accesses

Years and Authors of Summarized Original Work

  • 2006, 2008; Valiant

  • 2008, 2009, 2010, 2011; Cai, Lu

  • 2009; Cai, Choudhary, Lu

  • 2014; Cai, Gorenstein

Problem Definition

Holographic algorithm, introduced by L. Valiant [11], is an algorithm design technique rather than a single algorithm for a particular problem. In essence, these algorithms are reductions to the FKT algorithm [7–9] to count the number of perfect matchings in a planar graph in polynomial time. Computation in these algorithms is expressed and interpreted through a choice of linear basis vectors in an exponential “holographic” mix, and then it is carried out by the FKT method via the Holant Theorem. This methodology has produced polynomial time algorithms for a variety of problems ranging from restrictive versions of satisfiability, vertex cover, to other graph problems such as edge orientation and node/edge deletion. No polynomial time algorithms were known for these problems, and some minor variations are known to be NP-hard...

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Recommended Reading

  1. Cai JY, Gorenstein A (2014) Matchgates revisited. Theory Comput 10(7):167–197

    Article  MathSciNet  MATH  Google Scholar 

  2. Cai JY, Lu P (2008) Basis collapse in holographic algorithms. Comput Complex 17(2):254–281

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai JY, Lu P (2009) Holographic algorithms: the power of dimensionality resolved. Theor Comput Sci Comput Sci 410(18):1618–1628

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai JY, Lu P (2010) On symmetric signatures in holographic algorithms. Theory Comput Syst 46(3):398–415

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai JY, Lu P (2011) Holographic algorithms: from art to science. J Comput Syst Sci 77(1):41–61

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  6. Cai JY, Choudhary V, Lu P (2009) On the theory of matchgate computations. Theory Comput Syst 45(1):108–132

    Article  MathSciNet  MATH  Google Scholar 

  7. Kasteleyn PW (1961) The statistics of dimers on a lattice. Physica 27:1209–1225

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  8. Kasteleyn PW (1967) Graph theory and crystal physics. In: Harary F (ed) Graph theory and theoretical physics. Academic, London, pp 43–110

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  9. Temperley HNV, Fisher ME (1961) Dimer problem in statistical mechanics – an exact result. Philos Mag 6:1061–1063

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  10. Valiant LG (2006) Accidental algorthims. In: FOCS ’06: proceedings of the 47th annual IEEE symposium on foundations of computer science. IEEE Computer Society, Washington, pp 509–517. doi:http://dx.doi.org/10.1109/FOCS.2006.7

  11. Valiant LG (2008) Holographic algorithms. SIAM J Comput 37(5):1565–1594. doi:http://dx.doi.org/10.1137/070682575

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Author information

Authors and Affiliations

  1. Beijing University, Beijing, China

    Jin-Yi Cai

  2. Computer Sciences Department, University of Wisconsin–Madison, Madison, WI, USA

    Jin-Yi Cai

  3. Microsoft Research Asia, Shanghai, China

    Pinyan Lu

  4. The State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing, China

    Mingji Xia

Authors
  1. Jin-Yi Cai
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  2. Pinyan Lu
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  3. Mingji Xia
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Corresponding authors

Correspondence to Jin-Yi Cai , Pinyan Lu or Mingji Xia .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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© 2016 Springer Science+Business Media New York

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Cai, JY., Lu, P., Xia, M. (2016). Holographic Algorithms. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_746

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