Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

TL;DR

This paper introduces new natural polynomial families in algebraic complexity, demonstrating their intermediate status in the VNP class, and explores their relationships with well-known complexity classes and polynomials.

Contribution

It provides the first explicit examples of VNP-intermediate polynomials based on NP-complete problems, and analyzes their complexity properties over different fields.

Findings

New VNP-intermediate polynomial families based on NP-complete problems.

Over finite fields, these families are in VNP but not VNP-hard or in VP.

Certain polynomials based on satisfiability and Hamiltonian cycle are not monotone projections of the permanent.

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 B\"{u}rgisser 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 $\mathsf{VBP}$.