Improved Blocky Partition, Cover, and Equality-Rank Bounds for Set Disjointness via Kronecker Rebalancing
Year: 2026
Type: preprint
Abstract: A binary matrix $B\in\{0,1\}^{m\times n}$ is blocky (an equality matrix) if its $1$-entries form pairwise row- and column-disjoint rectangles. For Set Disjointness $D_d=D_1^{\otimes d}\in\{0,1\}^{2^d\times 2^d}$, Williams (FOCS 2024) bounded blocky cover number $\mathrm{bc}(D_d)$ and weak equality rank $\mathrm{weq}_{\mathbb{R}}(D_d)$ by $O(6^{d/6})\approx O(1.3480^d)$ and blocky partition number $\mathrm{bp}(D_d)$ and exact equality rank $\mathrm{br}_R(D_d)$ by $O(5^{d/5})\approx O(1.3797^d)$. While multi-bucket separator circuits give $\tilde{O}(n\cdot(2/\sqrt{3})^d)$-time and -space Orthogonal Vectors (Nederlof and W\k{e}grzycki, STOC 2021; Alman and Li, FOCS 2025) and unrestricted depth-$2$ circuits solve $\#\mathrm{OV}$ in $o(n\cdot 1.25^d)$ time (Alman and Li, 2025; Ye, 2026), $O(5^{d/5})$ and $O(6^{d/6})$ remained the best published bounds on $\mathrm{bp}(D_d)$ and $\mathrm{bc}(D_d)$ (with the exact cover base $2/\sqrt{3}$ established in a companion paper). First, via nonnegative LP dual vertex potentials we prove $\mathrm{fbc}(D_d)=\mathrm{bp}^*(D_d)$ for $1\le d\le 5$ with values $(3/2,7/4,23/10,35/13,205/62)$ (showing $\mathrm{bc}(D_5),\mathrm{bp}(D_5)\in\{4,5\}$, alongside $\mathrm{bp}(D_4)=4$); Kronecker submultiplicativity and greedy cover rounding yield $\mathrm{bc}(D_d)=O(d\cdot(205/62)^{d/5})=O(d\cdot 1.27021^d)$ (and the same bound on $\mathrm{weq}_{\mathbb{F}}(D_d)$ over every infinite or $>3^d$-sized field $\mathbb{F}$). Second, we introduce a palette Kronecker rebalancing LP $\tau_{\mathrm{bp}}(M)$ over exact rectangle partitions and blocky packings, proving $\tau_{\mathrm{bp}}(D_1,D_2,D_3)=(3/2,11/6,23/10)$, $\mathrm{bp}(A\otimes D_3)\le 276\lceil\mathrm{bp}(A)/120\rceil$ (refined to $92\lceil\mathrm{bp}(A)/40\rceil$), and $\mathrm{bp}(D_d)<3\cdot(23/10)^{d/3}<3\cdot 1.320007^d$ ($<11\cdot(23/10)^{d/3}$ from $120\to 276$ alone), which bounds $\mathrm{br}_R(D_d)$ ($R\ne 0$) and $\mathrm{spr}_{\mathbb{F}}(D_d),\mathrm{weq}_{\mathbb{F}}(D_d)$ over every field $\mathbb{F}$.
Author: Grey, Kevin
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225003055