Lecture 3 - Complete Polynomials II

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

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: $\mathrm{VNP}={\mathrm{VNP}}_{\mathrm{e}}$. 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 ${\mathrm{VP}}_{\mathrm{e}}$ such that $$f_n(x_1,\dots ,x_{t(n)})=\sum _{b\in {0,1}^{s(n)}}{g}_n(x_1,\dots ,x_{t(n)},b),$$ 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 $y_1,\dots ,y_n$ be $n$ new variables. Note that the Permanent is a “coefficient” of the following polynomial: $$p(X,y_1,\dots ,y_n):=\prod _{i\in [n]}(\sum _{j\in [n]}{X}_{ij}\cdot y_j).$$ 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 ${\mathrm{VNP}}_{\mathbb{F}}$.