Lecture 2 - Complete Polynomials I
Completeness of Determinant for $\mathrm{VBP}_{\mathbb{F}}$
We will show that the determinant is complete for $\mathrm{VBP}_{\mathbb{F}}$. This part is taken from Ramprasad Saptharishi’s lower bound survey, Section 3.3.3. To show membership of the determinant in $\mathrm{VBP}_{\mathbb{F}}$, we need the definition of a closed walk sequence (clow sequence).
Definition 1 (clow sequences): Let $G = ([n], E)$ be a directed graph with $n$ vertices. A closed walk (clow) of length $\ell$ in $G$ is a closed walk $W = (v_1, v_2, \ldots, v_{\ell}, v_1)$ such that $(v_i, v_{i+1}) \in E$ and $v_1 < v_i$ for all $i \in [\ell]$. We refer to $v_1$ as the head of the clow. In other words, the head of the clow is the smallest vertex in the walk, and the head does not repeat in the walk (intermediate vertices can repeat).
A closed walk sequence (clow sequence) is a sequence of clows $W_1, W_2, \ldots, W_r$ such that $head(W_i) < head(W_{i+1})$ for all $i \in [r-1]$. The length of a clow sequence is the sum of the lengths of the clows in the sequence. The weight of a clow sequence is the product of the weights of the edges in the clows in the sequence (with multiplicity). The sign of a clow sequence of length $\ell$ with $r$ clows is $(-1)^{\ell + r}$.
The definition of clow sequences generalizes the definition of cycle covers. In a beautiful result, Mahajan and Vinay showed that the determinant of a matrix can be expressed as a sign-weighted sum of clow sequences.
Lemma 1 (Mahajan and Vinay 1997): Let $X_G$ be the adjacency matrix of a directed weighted graph $G$. Then, $$\det(X_G) = \sum_{W \in \mathcal{W}_n} \text{sign}(W) \cdot \text{weight}(W),$$ where $\mathcal{W}_n$ is the set of clow sequences of length $n$ in $G$.
The proof of the above lemma is by showing that the set of clow sequences which are not cycle covers can be partitioned into pairs of clow sequences that cancel each other out.
Claim 1: the determinant is in $\mathrm{VBP}_{\mathbb{F}}$. More precisely, $\det_n(X)$ can be computed by an ABP of size $O(n^3)$.
Proof: Let us construct an ABP for computing the determinant, by computing a sign-weighted sum of clow sequences of length $n$ of the complete directed graph $\vec{K}_n$ on $n$ vertices.
The ABP will have $n+1$ layers, labeled $1, 2, \ldots, n+1$. Each layer $2 \leq \ell \leq n$ will have $n^2$ nodes, labeled $v^{(\ell)}_{ij}$, where $1 \leq i, j \leq n$. Think of index $i$ as representing the head of the current clow, and index $j$ as the current vertex in the clow. The length of the partial clow sequence constructed so far is $\ell$.
We will now prove the completeness of the determinant for $\mathrm{VBP}_{\mathbb{F}}$.
Claim 2: the determinant is complete for $\mathrm{VBP}_{\mathbb{F}}$.
Proof: Given any ABP $\Phi$ of size $r$ computing a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$, we will construct a square matrix $Y$ with entries being affine forms over $\mathbb{F}$ such that $\det(Y) = f$.
Determinants Simulate Algebraic Formulas
In this section we will prove that an algebraic formula of size $s$ can be simulated by a branching program of size $O(s)$. Thus, by the completeness of the determinant for $\mathrm{VBP}_{\mathbb{F}}$, we will have that determinants can efficiently simulate algebraic formulas.