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 $\tilde{\mathcal{O}}(1.35^d \cdot n)$-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 $\tilde{\mathcal{O}}(1.25^d n)$. 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.

翻译：在正交向量（OV）问题中，给定两个族 $A, B$，均为 $\{1,\ldots,d\}$ 的子集构成的族，每个族大小为 $n$，任务是判断是否存在一对 $a \in A$ 和 $b \in B$，使得 $a \cap b = \emptyset$。该问题的直接算法运行时间为 $\mathcal{O}(n^2 \cdot d)$ 或 $\mathcal{O}(2^d \cdot n)$，且假设SETH成立，则不存在 $2^{o(d)}\cdot n^{2-\varepsilon}$ 时间的算法能对任意常数 $\varepsilon > 0$ 解决该问题。Williams（FOCS 2024）基于不相交矩阵的简洁等秩分解，提出了一个 $\tilde{\mathcal{O}}(1.35^d \cdot n)$ 时间的算法。本文提出一个组合算法，其随机运行时间为 $\tilde{\mathcal{O}}(1.25^d n)$。通过计算机辅助评估，该结果可改进至 $\mathcal{O}(1.16^d \cdot n)$。我们将结果推广至 $k$-正交向量问题：给定 $k$ 个族 $A_1,\ldots,A_k$，均为 $\{1,\ldots,d\}$ 的子集构成的族，每个族大小为 $n$，任务是为每个 $i \in \{1,\ldots,k\}$ 寻找元素 $a_i \in A_i$，使得 $a_1 \cap a_2 \cap \ldots \cap a_k = \emptyset$。我们证明，对于任意固定的 $k \ge 2$，存在 $\varepsilon_k > 0$，使得 $k$-OV 问题可在 $\mathcal{O}(2^{(1 - \varepsilon_k)\cdot d}\cdot n)$ 时间内解决。我们还证明，渐进意义上这是最佳可能结果：对于任意 $\varepsilon > 0$，存在某个 $k \ge 2$，使得 $2^{(1 - \varepsilon)\cdot d} \cdot n^{\mathcal{O}(1)}$ 时间的 $k$-正交向量算法将推翻集合覆盖猜想。