New and Improved Concrete Lower Bounds for Orthogonal Vectors

2026-07-26

AI Summary

This work addresses the complexity of the Orthogonal Vectors (OV) problem and its k-ary generalization in several fundamental computational models, where unconditional concrete lower bounds have long been elusive. Focusing on monotone circuits, Boolean formulas, and branching programs, we unconditionally establish the monotone variants of both OV and the k-OV conjecture within these models. By integrating techniques from circuit complexity with an analysis of monotone computation, we prove that for dimension $d = c \log n$, any monotone circuit refuting OV requires size at least $\Omega(n^{2-\varepsilon})$. Additionally, we derive a tight Boolean formula lower bound of $\Omega(n^2 d)$, which is optimal up to constant factors. These results provide the first unconditional evidence supporting the hardness of OV in monotone settings across multiple classical models of computation.

Abstract

The Orthogonal Vectors Problem (OV$_{n,d}$) takes as input two sets $A,B$ each containing $n$ $d$-dimensional Boolean vectors, and outputs $1$ if and only if there exists $a \in A$ and $b \in B$ such that $a$ and $b$ are orthogonal. The OV conjecture states that for every $\varepsilon > 0$, there exists a constant $c \geq 1$ such that there is no algorithm deciding OV$_{n,d}$ for $d = c \log n$ with running time $O(n^{2-\varepsilon})$. The analogous $k$-OV conjecture hypothesizes a lower bound of $n^{k-ε}$ for the same problem with $k$ sets. We prove these results and variants unconditionally in concrete computational models. We study a natural monotone version of the $k$-OV conjecture and shows that it holds for monotone circuits and constant-depth (not necessarily monotone) circuits when $d = n^{Ω(1)}.$ We show that the monotone version of the OV conjecture holds for monotone circuits. More formally, we show that for every $ε> 0$, there exists $c$ such that any monotone circuit family computing the negation of OV$_{n,d}$ with $d=c\log n$ must have size $Ω(n^{2-ε})$. We also prove stronger Boolean formula and branching program lower bounds for OV$_{n,d}$, strengthening a previous result of Kane and Williams (ITCS 2019). In particular, our Boolean formula lower bound of $Ω(n^2 d)$ is tight up to constant factors.

Problem

• Orthogonal Vectors
• Computational Lower Bounds
• Monotone Circuits
• Boolean Formulas
• k-OV Conjecture

Innovation

• Orthogonal Vectors
• monotone circuits
• lower bounds
• Boolean formulas
• computational complexity