正交向量的新的和改进的具体下界
New and Improved Concrete Lower Bounds for Orthogonal Vectors
AI总结：
研究正交向量问题（OV$_{n,d}$）及其$k$-OV猜想，在具体计算模型中无条件证明相关结果及变体，研究其单调版本，证明对特定电路成立，还得出更强的布尔公式和分支程序下界，布尔公式下界在常数因子范围内紧。
AI中文摘要：
正交向量问题（OV$_{n,d}$）以两个集合$A$、$B$为输入，每个集合包含$n$个$d$维布尔向量，当且仅当存在$a \in A$和$b \in B$使得$a$和$b$正交时输出$1$。OV猜想指出，对于每个$\varepsilon > 0$，存在常数$c \geq 1$，使得对于$d = c \log n$，不存在运行时间为$O(n^{2 - \varepsilon})$的算法来判定OV$_{n,d}$。类似的$k$-OV猜想假设对于$k$个集合的相同问题有下界$n^{k - \epsilon}$。我们在具体计算模型中无条件地证明了这些结果及其变体。我们研究了$k$-OV猜想的自然单调版本，表明当$d = n^{\Omega(1)}$时，它对单调电路和常数深度（不一定单调）电路成立。我们还证明了关于OV$_{n,d}$更强的布尔公式和分支程序下界，加强了Kane和Williams（ITCS 2019）的先前结果。特别是，我们的布尔公式下界$\Omega(n^2 d)$在常数因子范围内是紧的。
英文摘要：
The Orthogonal Vectors Problem (OV$_{n,d}$) takes as input two sets $A,B$ each containing $n$ $d$-dimensional Boolean vectors, and outputs $1$ if and only if there exists $a \in A$ and $b \in B$ such that $a$ and $b$ are orthogonal. The OV conjecture states that for every $\varepsilon > 0$, there exists a constant $c \geq 1$ such that there is no algorithm deciding OV$_{n,d}$ for $d = c \log n$ with running time $O(n^{2-\varepsilon})$. The analogous $k$-OV conjecture hypothesizes a lower bound of $n^{k-ε}$ for the same problem with $k$ sets. We prove these results and variants unconditionally in concrete computational models. We study a natural monotone version of the $k$-OV conjecture and shows that it holds for monotone circuits and constant-depth (not necessarily monotone) circuits when $d = n^{Ω(1)}.$ We show that the monotone version of the OV conjecture holds for monotone circuits. More formally, we show that for every $ε> 0$, there exists $c$ such that any monotone circuit family computing the negation of OV$_{n,d}$ with $d=c\log n$ must have size $Ω(n^{2-ε})$. We also prove stronger Boolean formula and branching program lower bounds for OV$_{n,d}$, strengthening a previous result of Kane and Williams (ITCS 2019). In particular, our Boolean formula lower bound of $Ω(n^2 d)$ is tight up to constant factors.
