2026-09-08T02:22:06Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26474 2026-09-05T19:45:06Z ddc:004 open_access

Faster Algorithms for k-Orthogonal Vectors in Low Dimension Dürr, Anita Kipouridis, Evangelos Lampis, Michael Węgrzycki, Karol Orthogonal Vectors Fine-grained Complexity Exact Algorithms Set Cover In the Orthogonal Vectors problem (OV), we are given two families A, B of subsets of {1,…,d}, each of size n, and the task is to decide whether there exists a pair a ∈ A and b ∈ B such that a ∩ b = ∅. Straightforward algorithms for this problem run in 𝒪(n² ⋅ d) or 𝒪(2^d ⋅ n) time, and assuming SETH, there is no 2^o(d)⋅ n^{2-ε} time algorithm that solves this problem for any constant ε > 0. Williams (FOCS 2024) presented a 𝒪̃(1.35^d ⋅ n)-time algorithm for the problem, based on the succinct equality-rank decomposition of the disjointness matrix. In this paper, we present a combinatorial algorithm that runs in randomized time 𝒪̃(1.25^d ⋅ n). This can be improved to 𝒪(1.16^d ⋅ n) using computer-aided evaluations. We also consider a more general k-Orthogonal Vectors problem, where given k families A_1,…,A_k of subsets of {1,…,d}, each of size n, the task is to find elements a_i ∈ A_i for every i ∈ {1,…,k} such that a₁ ∩ a₂ ∩ … ∩ a_k = ∅. We show that for every fixed k ⩾ 2, there exists ε_k > 0 such that the k-OV problem can be solved in time 𝒪(2^{(1 - ε_k)⋅d} ⋅ n). We also show that, asymptotically, this is the best we can hope for: for any ε > 0 there exists a k ⩾ 2 such that 2^{(1 - ε)⋅ d} ⋅ n^𝒪(1) time algorithm for k-Orthogonal Vectors would contradict the Set Cover Conjecture. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Anita Dürr and Evangelos Kipouridis and Michael Lampis and Karol Węgrzycki 2026 Is Part Of LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2026.85 urn:nbn:de:0030-drops-264747 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.85 eng https://creativecommons.org/licenses/by/4.0/legalcode