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.

翻译：自Raz（《计算理论》，2006年）在超多项式多线性公式下界方面取得突破以来，证明多线性代数分支程序（mABP）的下界一直是代数复杂性理论中一个长期未决的开放问题。目前所有已知的多线性下界都依赖于最小划分秩方法，而针对mABP的最佳下界仍停留在二次阶（Alon、Kumar和Volk，《组合学》，2020年）。我们证明，最小划分秩方法无法证明超多项式mABP下界：存在一个可由多项式规模mABP计算的满秩多线性多项式。这是一个无条件障碍：要区分$\mathsf{mVBP}$与多线性层级中的更高类，需要新的技术。我们的证明解决了Fabris、Limaye、Srinivasan和Yehudayoff（ECCC，2026年）提出的一个开放问题，他们证明了该方法的效力受限于组合对象（称为1-平衡链集系统）的最小规模$N(n)$，并给出$N(n) \le n^{O(\log n/\log\log n)}$。我们通过赋予链构建者在每一步一个二元选择，将原本对称的随机游走偏置为不平衡增加概率不超过$1/4$的游走，从而证明$N(n) = n^{O(1)}$；结合一个鞅论证与多尺度递归，得到多项式界。