Lecture 3 - Complete Polynomials II

For this lecture, we will assume that the ground field $\mathbb{F}$ has $\mathrm{char}(\mathbb{F}) \neq 2$.

Moreover, as mentioned in the last lecture, we will also assume the following fact, which was proved by Valiant [Val 82], and proofs of this fact can be found in [B, ] and in [MP'08].

Theorem 1: $\mathrm{VNP} = \mathrm{VNP_e}$. That is, for any family of polynomials ${f_n}_{n \in \mathbb{N}}$ in $\mathrm{VNP}$, there exists a family of polynomials ${g_n}_{n \in \mathbb{N}}$ in $\mathrm{VP_e}$ such that $$ f_n(x_1, \ldots, x_{t(n)}) = \sum_{b \in {0,1}^{s(n)}} g_n(x_1, \ldots, x_{t(n)}, b),$$ where $s(n) = \mathrm{poly}(n)$ and $t(n) = \mathrm{poly}(n)$.

Moreover, we will also use the fact that any formula of size $s$ can be simulated by an ABP of size $O(s)$, as you will prove this in Homework 1.

Completeness of the Permanent for $\mathrm{VNP}_{\mathbb{F}} $

Claim 1: ${\mathrm{Per}_n }_{n \in \mathbb{N}} \in \mathrm{VNP}_{\mathbb{F}}$.

Proof: Let $X$ be an $n \times n$ symbolic matrix and $y_1, \ldots, y_n$ be $n$ new variables. Note that the Permanent is a “coefficient” of the following polynomial: $$ p(X, y_1, \dots, y_n) := \prod_{i \in [n]} \left( \sum_{j \in [n]} X_{ij} \cdot y_j \right).$$ where we consider $p$ as a polynomial in $y_1, \dots, y_n$ with coefficients in $\mathbb{F}[X]$. Moreover, it is easy to see that $p$ can be computed by a polynomial-size arithmetic formula.

$$ \mathrm{Per}_n(X) = \sum_{S \subset [n]} (-1)^{n-|S|} \cdot \prod_{i \in [n]} \left( \sum_{j \in S} X_{ij} \right) $$

We now come to the most challenging part of the lecture: completeness of the Permanent.

Claim 2: ${\mathrm{Per}_n }$ is complete for $\mathrm{VNP}_{\mathbb{F}}$.