An Unconditional Barrier for Proving Multilinear Algebraic Branching Program Lower Bounds

TL;DR

This paper establishes an unconditional barrier showing that the min-partition rank method cannot prove superpolynomial lower bounds for multilinear algebraic branching programs, highlighting the need for new techniques.

Contribution

It proves that the min-partition rank method cannot establish superpolynomial lower bounds for mABPs, resolving an open problem and demonstrating a polynomial bound on a key combinatorial parameter.

Findings

The min-partition rank method cannot prove superpolynomial mABP lower bounds.

A polynomial bound on the size of 1-balanced-chain set systems is established.

The proof introduces a novel chain-building technique with a biasing strategy.

Abstract

Since the breakthrough superpolynomial multilinear formula lower bounds of Raz (Theory of Computing 2006), proving such lower bounds against multilinear algebraic branching programs (mABPs) has been a longstanding open problem in algebraic complexity theory. All known multilinear lower bounds rely on the min-partition rank method, and the best bounds against mABPs have remained quadratic (Alon, Kumar, and Volk, Combinatorica 2020). We show that the min-partition rank method cannot prove superpolynomial mABP lower bounds: there exists a full-rank multilinear polynomial computable by a polynomial-size mABP. This is an unconditional barrier: new techniques are needed to separate $\mathsf{mVBP}$ from higher classes in the multilinear hierarchy. Our proof resolves an open problem of Fabris, Limaye, Srinivasan, and Yehudayoff (ECCC 2026), who showed that the power of this method is governed by the minimum size $N(n)$ of a combinatorial object called a $1$-balanced-chain set system, and proved $N(n) \le n^{O(\log n/\log\log n)}$. We prove $N(n) = n^{O(1)}$ by giving the chain-builder a binary choice at each step, biasing what was a symmetric random walk into one where the imbalance increases with probability at most $1/4$; a supermartingale argument combined with a multi-scale recursion yields the polynomial bound.