A Dichotomy Theorem for Homomorphism Polynomials
Computational Complexity
2012-10-30 v1
Abstract
In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by arithmetic circuits in constant depth if H has a loop or no edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by B\"urgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size).
Cite
@article{arxiv.1210.7641,
title = {A Dichotomy Theorem for Homomorphism Polynomials},
author = {Nicolas de Rugy-Altherre},
journal= {arXiv preprint arXiv:1210.7641},
year = {2012}
}