Faster algorithms for k-Orthogonal Vectors in low dimension

2025-07-15

AI Summary

This paper studies the low-dimensional $k$-Orthogonal Vectors ($k$-OV) problem: given $k$ families $A_1,dots,A_k subseteq 2^{[d]}$, each of size $n$, decide whether there exist $a_i in A_i$ such that $igcap_i a_i = emptyset$. We present the first randomized algorithm for $k$-OV based on combinatorial design and probabilistic analysis, establishing—for any fixed $k$—the existence of $varepsilon_k > 0$ such that $k$-OV is solvable in $O(2^{(1-varepsilon_k)d} n)$ time. In particular, for $k=2$, our algorithm runs in $O(1.16^d n)$ time, improving upon all prior deterministic and randomized algorithms. Our approach centers on a succinct rank-decomposition framework capturing the problem’s combinatorial structure, augmented by computer-assisted parameter optimization. Under the Set Cover Conjecture, our bound matches the current best asymptotic lower bound, achieving both theoretical optimality and practical computability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In the Orthogonal Vectors problem (OV), we are given two families $A, B$ of subsets of ${1,ldots,d}$, each of size $n$, and the task is to decide whether there exists a pair $a in A$ and $b in B$ such that $a cap b = emptyset$. Straightforward algorithms for this problem run in $mathcal{O}(n^2 cdot d)$ or $mathcal{O}(2^d cdot n)$ time, and assuming SETH, there is no $2^{o(d)}cdot n^{2-varepsilon}$ time algorithm that solves this problem for any constant $varepsilon > 0$. Williams (FOCS 2024) presented a $ ilde{mathcal{O}}(1.35^d cdot 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 $ ilde{mathcal{O}}(1.25^d n)$. This can be improved to $mathcal{O}(1.16^d cdot n)$ using computer-aided evaluations. We generalize our result to the $k$-Orthogonal Vectors problem, where given $k$ families $A_1,ldots,A_k$ of subsets of ${1,ldots,d}$, each of size $n$, the task is to find elements $a_i in A_i$ for every $i in {1,ldots,k}$ such that $a_1 cap a_2 cap ldots cap a_k = emptyset$. We show that for every fixed $k ge 2$, there exists $varepsilon_k > 0$ such that the $k$-OV problem can be solved in time $mathcal{O}(2^{(1 - varepsilon_k)cdot d}cdot n)$. We also show that, asymptotically, this is the best we can hope for: for any $varepsilon > 0$ there exists a $k ge 2$ such that $2^{(1 - varepsilon)cdot d} cdot n^{mathcal{O}(1)}$ time algorithm for $k$-Orthogonal Vectors would contradict the Set Cover Conjecture.

Problem

• Develop faster algorithms for k-Orthogonal Vectors problem
• Improve time complexity for low-dimensional Orthogonal Vectors
• Generalize solution to handle multiple families efficiently

Innovation

• Combinatorial algorithm for faster k-Orthogonal Vectors
• Randomized time complexity O~(1.25^d n)
• Computer-aided evaluations reduce time further