An Unconditional Barrier for Proving Multilinear Algebraic Branching Program Lower Bounds

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.

Figures (2)

• Figure 1: An example of a steered path on the two-interval grid with $m/2 = 7$. The $\pm 1$ labels along each axis show the $f$-values of the elements of $I^L$ (horizontal) and $I^R$ (vertical); the numbers inside each node show the running imbalance $H(t)$; let $\hat{h}(t) = |H(t)|$ denote the absolute imbalance. The path begins at $(0,0)$ with $H = 0$. At step 1, both options give $\hat{h} = 1$, so a coin flip decides (dashed); the coin selects right, giving $H = {+1}$. At step 4, from $(1,2)$ with $H = {+1}$: extending $I^L$ adds $f = {-1}$ (reducing $\hat{h}$ to $0$), while extending $I^R$ adds $f = {+1}$ (increasing $\hat{h}$ to $2$); the builder picks the reducing option (solid). At step 6 ($\bigstar$), from $(3,2)$ with $H = {+1}$: both candidates have $f = {+1}$, so $\hat{h}$ must increase to $2$ regardless of the builder's choice---this is a "bad" event whose probability is bounded in Section \ref{['sec:probability']}. At the final step, $I^L$ is exhausted and only $I^R$ remains (dotted).
• Figure 2: A schematic trajectory of the absolute imbalance $\hat{h}(t) = |H(t)|$ along the steered path within a single scale of the construction. The shaded band marks the balanced region$\hat{h} \leq 1$: when the trajectory is in this region, the chain set $C_t$ has $|f(C_t)| \leq 1$. The dashed line marks the height bound $1 + K\ln m$ (Theorem \ref{['thm:supermg']}): the trajectory stays below this bound with high probability. Each departure from and return to the balanced region constitutes an excursion; a gap is the interval between two consecutive balanced visits. By Corollary \ref{['cor:gap']}, every gap has length at most $C_1\ln m$. During each gap, the gap-filling lemma (Lemma \ref{['lem:greedy']}) provides a $1$-balanced ordering of the gap elements, and the set system includes all subsets of each gap region to accommodate every possible such ordering. The key qualitative features are that the trajectory returns to the balanced region frequently and never strays far from it---both consequences of the negative bias ($p \leq 1/4$). Compare this with the approach of Fabris et al. (Section \ref{['sec:flsy-approach']}), where the imbalance follows a symmetric random walk: there, the maximum height is $\Theta(\sqrt{m})$ and the gaps between returns are $\Theta(m/\log m)$, necessitating a deep recursion.

Theorems & Definitions (44)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3: Main result
• Corollary 1.4
• Remark 1.5
• Definition 2.1: Set systems and maximal chains
• Definition 2.2: Balanced colorings and chain-balance
• Definition 2.3: Average-case version
• Theorem 2.4: Worst-case to average-case, FLSY26
• Definition 3.1: Steered path