Precise boundary between Σπ set-multilinear ABPs and smVBP, and reachability of the escalation threshold by lopsided methods

Characterize the precise boundary between the class of sums of ordered set-multilinear algebraic branching programs (Σπ smABP) and the full set-multilinear ABP class smVBP, and ascertain whether the lopsided partial derivative method or related techniques can prove superpolynomial Σπ smABP lower bounds up to the O(log n / log log n) degree escalation threshold that would lift to general ABP lower bounds.

Background

The Σπ smABP model consists of sums of ordered set-multilinear ABPs, possibly with different variable orderings, and is known to be strictly weaker than smVBP by superpolynomial separations. Results of Bhargav, Dwivedi, and Saxena show that achieving superpolynomial lower bounds for Σπ smABPs at degree r ≤ O(log n / log log n) would escalate to superpolynomial lower bounds for general ABPs. Current lower bounds (e.g., CKSS24) achieve separations only for higher degrees. Determining the exact degree threshold separating Σπ smABP from smVBP and whether lopsided partial derivative techniques can meet the escalation threshold would clarify the prospects for lifting set-multilinear lower bounds to general ABP lower bounds.

References

The results of show that the $\Sigma_\pi$smABP model is strictly weaker than $\mathsf{smVBP}$ (superpolynomially so, even for polynomials with superlogarithmic degree in $\mathsf{smVBP}$). The results of show that proving $\Sigma_\pi$smABP lower bounds at slightly smaller degree would have dramatic consequences. Understanding the precise boundary—and whether the lopsided method or other techniques can reach the escalation threshold—is a compelling open problem.

— An Unconditional Barrier for Proving Multilinear Algebraic Branching Program Lower Bounds  (2604.00746 - Kush, 1 Apr 2026) in Section 7, Conclusion and open questions (item 2)