A Dichotomy Theorem for Homomorphism Polynomials

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


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).


Bipartite Graph Dichotomy Theorem Complete Bipartite Graph Homogeneous Component Conjunctive Query 
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© 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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