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A Complete Dichotomy for Complex-Valued Holant^c

Author Miriam Backens

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Author Details

Miriam Backens
  • Department of Computer Science, University of Oxford, UK

Funding

  • Backens, Miriam: The research leading to these results has received funding from EPSRC via grant EP/L021005/1 and from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the author’s views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein. No new data were created during this study. The majority of this research was done at the School of Mathematics, University of Bristol, UK.

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Miriam Backens. A Complete Dichotomy for Complex-Valued Holant^c. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.12

Abstract

Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions, i.e. functions which only depend on the Hamming weight of the input.
Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement. This full dichotomy for Holant^c answers a question that has been open for almost a decade.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • computational complexity
  • counting complexity
  • Holant problems
  • dichotomy
  • entanglement

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References

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