eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-03-11
25:1
25:17
10.4230/LIPIcs.STACS.2024.25
article
Depth-3 Circuit Lower Bounds for k-OV
Choudhury, Tameem
1
https://orcid.org/0000-0002-5044-9717
Sreenivasaiah, Karteek
1
https://orcid.org/0000-0001-7396-3383
Department of Computer Science and Engineering, IIT Hyderabad, India
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol289-stacs2024/LIPIcs.STACS.2024.25/LIPIcs.STACS.2024.25.pdf
fine grained complexity
k-OV
circuit lower bounds
depth-3 circuits