The Orthogonal Vectors Conjecture and Non-Uniform Circuit Lower Bounds
Ryan Williams · 2024
A line of work has shown how nontrivial uniform algorithms for analyzing circuits can be used to derive non-uniform circuit lower bounds. We show how the non-existence of nontrivial circuit-analysis algorithms can also imply non-uniform circuit lower bounds. Our connections yield new win-win circuit lower bounds, and suggest a potential approach to refuting the Orthogonal Vectors Conjecture in the$O(\log n)$-dimensional case, which would be sufficient for refuting the Strong Exponential Time Hypothesis (SETH). For example, we show that at least one of the following holds: • There is an$\varepsilon>0$such that for infinitely many$n$, read-once 2-DNFs on$n$variables cannot be simulated by non-uniform$2^{\varepsilon n}$-size depth-two exact threshold circuits. It is already a notorious open problem to prove that the class$E^{N P}$does not have polynomial-size depth-two exact threshold circuits, so such a lower bound would be a significant advance in low-depth circuit complexity. In fact, a stronger lower bound holds in this case: the$2^n \times 2^n$Disjointness Matrix (well-studied in communication complexity) cannot be expressed by a linear combination of$2^{o(n)}$structured matrices that we call “equality matrices”. • For every$c \geq 1$and every$\varepsilon>0$, Orthogonal Vectors on$n$vectors in$c \log n$dimensions can be solved in$n^{1+\varepsilon}$uniform deterministic time. This case would provide a strong refutation of the Orthogonal Vectors conjecture, and of SETH: for example, CNF-SAT on$n$variables and$O(n)$clauses could be solved in$2^{n / 2+o(n)}$time. Moreover, this case would imply non-uniform circuit lower bounds for the class$E^{NP}$, against Valiant series-parallel circuits. Inspired by this connection, we give evidence from SAT/SMT solvers that the first item (in particular, the Disjointness lower bound) may be false in its full generality. In particular, we present a systematic approach to solving Orthogonal Vectors via constant-sized decompositions of the Disjointness Matrix, which already yields interesting new algorithms. For example, using a linear combination of 6 equality matrices that express$2^6 \times 2^6$Disjointness, we derive an$\tilde{O}\left(n \cdot 6^{d / 6}\right) \leq \tilde{O}\left(n \cdot 1. \dot{35^d}\right)$time and$n \cdot \operatorname{poly}(\log n, d)$space algorithm for Orthogonal Vectors on$n$vectors in$d$dimensions. We show similar results for counting pairs of orthogonal vectors.