Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

  • Meena Mahajan
  • Nitin SaurabhEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


We provide a list of new natural \(\mathsf {VNP}\)-Intermediate polynomial families, based on basic (combinatorial) \(\mathsf {NP}\)-Complete problems that are complete under parsimonious reductions. Over finite fields, these families are in \(\mathsf {VNP}\), and under the plausible hypothesis \(\mathsf {Mod}_p\mathsf {P}\not \subseteq \mathsf {P/poly}\), are neither \(\mathsf {VNP}\)-hard (even under oracle-circuit reductions) nor in \(\mathsf {VP}\). Prior to this, only the Cut Enumerator polynomial was known to be \(\mathsf {VNP}\)-intermediate, as shown by Bürgisser in 2000.

We next show that over rationals and reals, two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. This augments recent results along this line due to Grochow.

Finally, we describe a (somewhat natural) polynomial defined independent of a computation model, and show that it is \(\mathsf {VP}\)-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established \(\mathsf {VP}\)-completeness of a related polynomial but under constant-depth oracle circuit reductions. Both polynomials are based on graph homomorphisms. A simple restriction yields a family similarly complete for \(\mathsf {VBP}\).


Hamiltonian Cycle Tree Decomposition Arithmetic Circuit Graph Homomorphism Affine Projection 
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 International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.The Institute of Mathematical SciencesChennaiIndia

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