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Depth-3 Circuit Lower Bounds for k-OV

Authors: Tameem Choudhury and Karteek Sreenivasaiah

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


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.

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)


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@InProceedings{choudhury_et_al:LIPIcs.STACS.2024.25,
  author =	{Choudhury, Tameem and Sreenivasaiah, Karteek},
  title =	{{Depth-3 Circuit Lower Bounds for k-OV}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.25},
  URN =		{urn:nbn:de:0030-drops-197359},
  doi =		{10.4230/LIPIcs.STACS.2024.25},
  annote =	{Keywords: fine grained complexity, k-OV, circuit lower bounds, depth-3 circuits}
}
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