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# Depth-3 Circuit Lower Bounds for k-OV

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LIPIcs.STACS.2024.25.pdf
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## Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments that helped improve the presentation of this paper in several respects.

## Cite As

Tameem Choudhury and Karteek Sreenivasaiah. Depth-3 Circuit Lower Bounds for k-OV. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.25

## Abstract

The 2-Orthogonal Vectors (2-OV) problem is the following: given two tuples A and B of n Boolean vectors, each of dimension d, decide if there exist vectors u ∈ A, and v ∈ B, such that u and v are orthogonal. This problem, and its generalization k-OV defined analogously for k tuples, are central problems in the area of fine-grained complexity. One of the major conjectures in fine-grained complexity is that k-OV cannot be solved by a randomised algorithm in n^{k-ε}poly(d) time for any constant ε > 0. In this paper, we are interested in unconditional lower bounds against k-OV, but for weaker models of computation than the general Turing Machine. In particular, we are interested in circuit lower bounds to computing k-OV by Boolean circuit families of depth 3 of the form OR-AND-OR, or equivalently, a disjunction of CNFs. We show that for all k ≤ d, any disjunction of t-CNFs computing k-OV requires size Ω((n/t)^k). In particular, when k is a constant, any disjunction of k-CNFs computing k-OV needs to use Ω(n^k) CNFs. This matches the brute-force construction, and for each fixed k > 2, this is the first unconditional Ω(n^k) lower bound against k-OV for a computation model that can compute it in size O(n^k). Our results partially resolve a conjecture by Kane and Williams [Daniel M. Kane and Richard Ryan Williams, 2019] (page 12, conjecture 10) about depth-3 AC⁰ circuits computing 2-OV. As a secondary result, we show an exponential lower bound on the size of AND∘OR∘AND circuits computing 2-OV when d is very large. Since 2-OV reduces to k-OV by projections trivially, this lower bound works against k-OV as well.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Circuit complexity
• Theory of computation → Problems, reductions and completeness
##### Keywords
• fine grained complexity
• k-OV
• circuit lower bounds
• depth-3 circuits

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