The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant of a matrix of polynomially bounded size. Strikingly, this restatement does not mention any notion of computational model. To get a similar restatement for the original and more fundamental question, and also to better understand the class itself, we need a complete polynomial for VP. Ad hoc constructions yielding complete polynomials were known, but not natural examples in the vein of the determinant. We give here several variants of natural complete polynomials for VP, based on the notion of graph homomorphism polynomials.

Homomorphism Polynomials Complete for VP.

We provide a list of new natural V
 N
 P-intermediate polynomial families, based on basic (combinatorial) N
 P-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in V
 N
 P, and under the plausible hypothesis M
 o
 d
 p
 P ⫅ P
 /
 p
 o
 l
 y, are neither V
 N
 P-hard (even under oracle-circuit reductions) nor in V
 P. Prior to this, only the Cut Enumerator polynomial was known to be V
 N
 P-intermediate, as shown by Burgisser in 2000. We show next that over rationals and reals, the clique polynomial cannot be obtained as a monotone p-projection of the permanent polynomial, thus ruling out the possibility of transferring monotone clique lower bounds to the permanent. We also show that two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. These results augment 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 V
 P-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established V
 P-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 V
 B
 P.

Some Complete and Intermediate Polynomials in Algebraic Complexity Theory.

In the uniform circuit model of computation, the width of a boolean circuit exactly characterizes the “space” complexity of the computed function. Looking for a similar relationship in Valiant’s algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width. We introduce the class VL as an algebraic variant of deterministic log-space L; VL is a subclass of VP.Further, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of “read-once” certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs (an algebraic analog of NL) can be expressed as read-once exponential sums over polynomials in $${{\sf VL}, {\it i.e.}\quad{\sf VBP} \in \Sigma^R \cdot {\sf VL}}$$ . Thus, read-once exponential sums can be viewed as a reasonable way of capturing space-bounded non-determinism. We also show that ΣR ·VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Though the best upper bound we have for ΣR ·VL itself is VNP, we can obtain better upper bounds for width-bounded multiplicatively disjoint (md-) circuits. Without the width restriction, md- arithmetic circuits are known to capture all of VP. We show that read-once exponential sums over md- constant-width arithmetic circuits are within VP and that read-once exponential sums over md- polylog-width arithmetic circuits are within VQP.We also show that exponential sums of a skew formula cannot represent the determinant polynomial.

Small space analogues of Valiant's classes and the limitations of skew formula.

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 $\mathsf{VBP}$.

Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

/pdf/homomorphism-polynomials-complete-for-vp-lnz4egabp7.pdf

Homomorphism Polynomials Complete for VP

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 Burgisser 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).

A dichotomy theorem for homomorphism polynomials

The fermionant ${\rm Ferm}^k_n(\bar x) = \sum_{\sigma \in S_n} (-k)^{c(\pi)}\prod_{i=1}^n x_{i,j}$ can be seen as a generalization of both the permanent (for k = − 1) and the determinant (for k = 1). We demonstrate that it is $\textsc{VNP}$-complete for any rational k ≠ 1. Furthermore it is #P-complete for the same values of k. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (when the Young diagram is a column). We demonstrate that the immanant of any family of Young diagrams with bounded width and at least n e boxes at the right of the first column is $\textsc{VNP}$-complete.

Determinant versus Permanent: Salvation via Generalization?

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 edge and that it is hard otherwise (i.e., complete for VNP, the arithmetic class related to #P). We also demonstrate the hardness over the rational field of cut eliminator, a polynomial defined by Burgisser which is known to be neither VP nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is the class of polynomials computable by arithmetic circuit of polynomial size).

A Dichotomy Theorem for Homomorphism Polynomials

The fermionant can be seen as a generalization of both the permanent (for $k=-1$) and the determinant. We demonstrate that it is VNP-complete for most cases. Furthermore it is #P-complete for the cases. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (when the Young diagram is a column). We demonstrate that the immanant of any family of Young diagrams with bounded width and at least n boxes at the right of the first column is VNP-complete.

Nicolas de Rugy-Altherre

Papers

A dichotomy theorem for homomorphism polynomials

Determinant versus Permanent: Salvation via Generalization?

A Dichotomy Theorem for Homomorphism Polynomials

Determinant versus Permanent: salvation via generalization? The algebraic complexity of the Fermionant and the Immanant.