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Homomorphism Polynomials Complete for VP

Authors: Arnaud Durand, Meena Mahajan, Guillaume Malod, Nicolas de Rugy-Altherre, and Nitin Saurabh

Published in: LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)


Abstract
The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant of a matrix of polynomially bounded size. Strikingly, this restatement does not mention any notion of computational model. To get a similar restatement for the original and more fundamental question, and also to better understand the class itself, we need a complete polynomial for VP. Ad hoc constructions yielding complete polynomials were known, but not natural examples in the vein of the determinant. We give here several variants of natural complete polynomials for VP, based on the notion of graph homomorphism polynomials.

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Arnaud Durand, Meena Mahajan, Guillaume Malod, Nicolas de Rugy-Altherre, and Nitin Saurabh. Homomorphism Polynomials Complete for VP. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 493-504, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{durand_et_al:LIPIcs.FSTTCS.2014.493,
  author =	{Durand, Arnaud and Mahajan, Meena and Malod, Guillaume and de Rugy-Altherre, Nicolas and Saurabh, Nitin},
  title =	{{Homomorphism Polynomials Complete for VP}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{493--504},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Raman, Venkatesh and Suresh, S. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.493},
  URN =		{urn:nbn:de:0030-drops-48665},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.493},
  annote =	{Keywords: algebraic complexity, graph homomorphism, polynomials, VP, VNP, completeness}
}