A Dichotomy Theorem for Homomorphism Polynomials

  • Nicolas de Rugy-Altherre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

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 edges and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over ℚ of cut eliminator, a polynomial defined by Bürgisser which is known to be neither VP nor VNP-complete in \(\mathbb F_2\), if VP ≠ VNP (VP is the class of polynomials computable by arithmetic circuits of polynomial size).

Keywords

Bipartite Graph Dichotomy Theorem Complete Bipartite Graph Homogeneous Component Conjunctive Query 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nicolas de Rugy-Altherre
    • 1
  1. 1.Institut de Mathématiques de Jussieu, UMR 7586 CNRSUniv Paris Diderot, Sorbonne Paris CitéParisFrance

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