eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2014-12-12
493
504
10.4230/LIPIcs.FSTTCS.2014.493
article
Homomorphism Polynomials Complete for VP
Durand, Arnaud
Mahajan, Meena
Malod, Guillaume
de Rugy-Altherre, Nicolas
Saurabh, Nitin
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol029-fsttcs2014/LIPIcs.FSTTCS.2014.493/LIPIcs.FSTTCS.2014.493.pdf
algebraic complexity
graph homomorphism
polynomials
VP
VNP
completeness