# On algebraic branching programs of small width

## TL;DR

This paper explores the algebraic complexity of small-width algebraic branching programs (ABPs), characterizes the topological closure of VP_e, and establishes new equivalences and separations among algebraic complexity classes.

## Contribution

It extends the equivalence between formula size and width-3 ABPs to approximate formulas and width-2 ABPs, and characterizes the closure VP_e-bar with a Fibonacci-like polynomial.

## Key findings

- Approximate formula size is polynomially equivalent to approximate width-2 ABP size.
- VP_e-bar can be characterized by a Fibonacci-like polynomial.
- VNP admits a hypercube summation representation, leading to class separations.

## Abstract

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the problem of separating these classes we study the topological closure VP_e-bar, i.e. the class of polynomials that can be approximated arbitrarily closely by polynomials in VP_e. We describe VP_e-bar with a strikingly simple complete polynomial (in characteristic different from 2) whose recursive definition is similar to the Fibonacci numbers. Further understanding this polynomial seems to be a promising route to new formula lower bounds. Our methods are rooted in the study of ABPs of small constant width. In 1992 Ben-Or and Cleve showed that formula size is polynomially equivalent to width-3 ABP size. We extend their result (in characteristic different from 2) by showing that approximate formula size is polynomially equivalent to approximate width-2 ABP size. This is surprising because in 2011 Allender and Wang gave explicit polynomials that cannot be computed by width-2 ABPs at all! The details of our construction lead to the aforementioned characterization of VP_e-bar. As a natural continuation of this work we prove that the class VNP can be described as the class of families that admit a hypercube summation of polynomially bounded dimension over a product of polynomially many affine linear forms. This gives the first separations of algebraic complexity classes from their nondeterministic analogs.

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Source: https://tomesphere.com/paper/1702.05328