Polynomial-Sized Algebraic Branching Programs

• Polynomially sized ABPs are directed acyclic graphs that compute polynomials via sums of path products, serving as a foundational model in algebraic complexity theory.
• They feature strong lower bounds, including quadratic size for power sums and exponential width for restricted read patterns, derived through algebraic and geometric techniques.
• Their simulation power bridges key complexity classes like VP and VNP, driving advances in efficient identity testing, derandomization, and model separation.

Polynomially sized algebraic branching programs (ABPs) are a central construct in algebraic complexity theory, capturing the class of polynomials computable by directed acyclic graphs whose paths encode algebraic computations. Their structure, power, and limitations connect fundamental complexity classes (such as VP, VNP, and subclasses of weakly-skew and formula models) and form the technical backbone for numerous recent advances in lower bounds, efficient identity testing, derandomization, and geometric complexity. This article surveys the formal models, foundational results, state-of-the-art lower bounds, algorithmic properties, and broader complexity-theoretic significance of polynomially sized ABPs and their principal variants.

1. Formal Models and Definitions

Algebraic branching programs are defined as follows. An ABP over field $\mathbb{F}$ is a directed acyclic (optionally layered) graph $G = (V, E)$, with a unique source $s$ and sink $t$. Each edge $e = (u \rightarrow v)$ is labeled by an affine or low-degree polynomial $\ell_e(\vec{x}) \in \mathbb{F}[x_1, ..., x_n]$. The program computes the polynomial

$$P(\vec{x}) = \sum_{\pi: s \rightarrow t} \prod_{e \in \pi} \ell_e(\vec{x}),$$

where the sum is over all $s$–$t$ paths.

The ABP’s size is generally measured as $|V|$, the total number of vertices, which up to polynomial factors coincides with the number of edges. Special subclasses are characterized by further structural or resource constraints:

• Width $w$ ABPs: Layered ABPs where each layer contains at most $w$ vertices; size is $O(w \cdot \ell)$ for length $\ell$.
• Read-once oblivious ABPs (ROABPs): Layered, width-bounded ABPs where each layer reads exactly one variable (in a fixed order), and each variable appears in exactly one layer.
• Read-$k$ oblivious ABPs: Each variable appears in at most $k$ layers.
• Homogeneous ABPs: On every $s$–$v$ path, the sum of degrees equals a fixed "formal degree"; in particular, every $s$–$t$ path has degree matching $\deg(P)$.

Classic complexity class correspondences:

• VP$_{\text{ws}}$ (weakly-skew circuits): polynomials computed by poly-size ABPs (Mengel, 2013).
• VP$_{\text{e}}$ (formulas): polynomials computable by arithmetic formulas; polynomially equivalent to poly-size, constant-width (width-3) ABPs for characteristic $\neq 2$ (Bringmann et al., 2017).
• VNP: poly-size ABPs with random-access memory (Mengel, 2013).

2. Quadratic and Exponential Lower Bounds

The most profound results delineate the limitations of polynomial-size ABPs.

Quadratic Lower Bound for Power Sums: For the $n$-variate polynomial

$f_n(x_1, ..., x_n) = \sum_{i=1}^n x_i^n,$

every ABP (over $\mathrm{char}(\mathbb{F}) \nmid n$) computing $f_n$ must have size $\Omega(n^2)$ (Chatterjee et al., 2019). The result strengthens the decades-long $\Omega(n \log n)$ barrier (Baur–Strassen argument) and demonstrates, for edge-labels of maximum degree $\Delta$, a lower bound of $\Omega(n^2 / \Delta)$.

The proof involves depth reduction to a shallow homogeneous ABP, which, up to controlled bilinear “error-terms,” computes $f_n$, and then a robust geometric lower bound based on the singular locus of the hypersurface defined by $f_n$. This establishes $\Omega(n^2)$ as an essentially tight lower bound for ABP size for these polynomials.

Exponential Width Lower Bounds in Restricted Models: For read-$k$ oblivious ABPs, there exists an explicit multilinear polynomial (computable by a poly-size depth-3 circuit) such that any read-$k$ oblivious ABP of width $w$ computing it must have $w \geq \exp(\Omega(n / k^{O(k)}))$, so polynomially sized ABPs must have super-polynomial width for restricted read patterns (Anderson et al., 2015). In the case of ROABPs ($k=1$), earlier results established exponential width lower bounds for certain explicit functions (Forbes et al., 2012).

Syntactic Multilinear ABPs: For sums of ROABPs, $r$-pass multilinear ABPs, and $\alpha$-sparse ROABPs, the best explicit polynomials require exponentially many summands or super-polynomial size, even though each summand is individually polynomial-size in the restricted model (Ramya et al., 2018).

3. Simulation Power and Class Separations

Ben-Or–Cleve Equivalence: For characteristic $\neq 2$, general formulas (VP$_{\mathrm{e}}$) and width-3, poly-size ABPs are polynomially equivalent—in formulas, every gate can be simulated by a width-3 product of $3 \times 3$ matrices, and vice versa (Bringmann et al., 2017).

Border Complexity – Width-2: All families of polynomials with polynomial border formula size can be arbitrarily well-approximated (in the $\varepsilon \to 0$ limit) by width-2 ABPs of polynomial size, yielding polynomial equivalence in the "border" sense: $$\overline{\mathrm{VP}_e} = \overline{\#_2\mathrm{VP}}$$ (Bringmann et al., 2017). Remarkably, this holds despite explicit polynomials not being exactly computable by width-2 ABPs.

Memory-Enhanced ABPs: By adding a stack, one obtains poly-size stack-ABPs characterizing VP; by adding random-access memory, the model captures VNP (Mengel, 2013). This yields the correspondences: | Model | Characterized Class | |----------------|---------------------------| | ABPs | VP$_{\mathrm{ws}}$ | | Stack-ABPs | VP | | RAM-ABPs | VNP |

4. Identity Testing and Derandomization

Efficient polynomial identity testing (PIT) for polynomial-size ABPs is a central derandomization challenge.

• General ABPs: PIT remains wide open; no polynomial or quasipolynomial-time algorithm is known (Anderson et al., 2015, Forbes et al., 2012).
• ROABPs: Quasipolynomial-size hitting sets exist when the variable order is known, with explicit hitting sets of size $\exp(O(\log^2 S))$ for circuits of size $S$ (Forbes et al., 2012). This algebraically mirrors Nisan’s pseudorandom generator for read-once Boolean branching programs.
• Read-$k$ ABPs: White-box PIT runs in time $2^{\widetilde{O}(n^{1 - 1/2^{k-1}})}$ for known read order (Anderson et al., 2015).
• $\pi$-Ordered/Oblivious ABPs: Deterministic black-box PIT for read-$r$ $\pi$-OABPs operates in $2^{O(r \log r \log^2 n \log\log n)}$ time via recursive explicit generators (Jansen et al., 2010).

The quasi-polynomial barrier for black-box derandomization reflects a possible "natural proofs" limitation: breaking this barrier may require qualitatively new techniques (Forbes et al., 2012).

5. Structural Properties and Separation Phenomena

Layered vs. Unlayered ABPs: Unlayered ABPs may be exponentially more efficient than their layered counterparts. There are explicit families that admit $O(n)$-size unlayered ABPs but require size at least $\Omega(n \sqrt{n})$ for any layered ABP, with an even sharper separation ($\Omega(n \log n)$) in the non-commutative setting (Engels, 2020).

Order-Finding and Hardness in ROABPs: Finding or even approximating the optimal variable order for minimal-width ROABPs is NP-hard, as is testing affine equivalence to a small-width ROABP, even for homogeneous or sparse polynomials, and even for quadratic forms (Ramya et al., 16 Sep 2025). These hardness results explain why many learning and PIT algorithms must assume fixed variable order, special polynomial structure, or restrict to subexponential time.

Non-commutative Separations: In structured non-commutative models, abecedarian ABPs of size $O(nd)$ can compute explicit abecedarian polynomials not computable by abecedarian formulas of polynomial size, exhibiting separations specific to the non-commutative subclass (Chatterjee, 2021).

6. Geometric Techniques and Refinements of Lower Bounds

The strongest lower bounds on ABP size arise from algebraic geometry. For homogeneous polynomials, the singular locus and the absence of low-codimension, low-degree subvarieties supplies precise lower bounds via "degree-restricted strength decompositions" (Gesmundo et al., 2022). For instance, for power-sum and Shioda hypersurfaces, classical intersection theory, Noether–Lefschetz type conditions, and slice-rank methods yield lower bounds matching explicit upper bounds in many cases.

The robustness of singular-locus based bounds enables extension from ABPs to general depth and to formulas computing symmetric polynomials, achieving quadratic lower bounds and surpassing previous $n^2/\log n$ limitations for formulas (Chatterjee et al., 2019).

7. Connections to Determinantal Complexity and Open Problems

Efficient transformations exist: every homogeneous polynomial of determinantal complexity $s$ can be computed by a homogeneous ABP of size $O(d^5 s)$, and for most polynomials, width $s - 1$ and size $O(ds)$ suffice (Chatterjee et al., 2023). Thus, for constant degree $d$, determinantal complexity and homogeneous ABP complexity are within a constant factor. Any super-linear lower bound for ABPs of constant degree immediately implies a super-linear lower bound for determinantal complexity, linking two major open problems.

Open directions and barriers include:

• Whether $\Omega(n^2)$ lower bounds for unlayered ABPs can be further improved (Engels, 2020).
• Extending approach beyond border complexity for small-width ABPs (Bringmann et al., 2017).
• Achieving polynomial-size black-box PIT for general ABPs or surmounting the exponential width barrier for restricted models (Anderson et al., 2015, Forbes et al., 2012).
• Developing geometric obstructions for stronger lower bounds or making progress on the separation between ABPs and circuits/formulas.

The paper of polynomially sized ABPs thus sits at a central crossroads of algebraic complexity, derandomization, and geometric methods, providing both technical depth and broad structural insights.