Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

Abstract

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 $\mathsf {VBP}$.