The problem of learning or reconstructing an unknown graph from a known family via partial-information queries arises as a mathematical model in various contexts. The most basic type of access to the graph is via \emph{edge queries}, where an algorithm may query the presence/absence of an edge between a pair of vertices of its choosing, at unit cost. While more powerful query models have been extensively studied in the context of graph reconstruction, the basic model of edge queries seems to have not attracted as much attention. In this paper we study the edge query complexity of learning a hidden bipartite graph, or equivalently its bipartite adjacency matrix, in the classical as well as quantum settings. We focus on learning matchings and half graphs, which are graphs whose bipartite adjacency matrices are a row/column permutation of the identity matrix and the lower triangular matrix with all entries on and below the principal diagonal being 1, respectively. \begin{itemize} \item For matchings of size $n$, we show a tight deterministic bound of $n(n-1)/2$ and an asymptotically tight randomized bound of $\Theta(n^2)$. A quantum bound of $\Theta(n^{1.5})$ was shown in a recent work of van Apeldoorn et al.~[ICALP'21]. \item For half graphs whose bipartite adjacency matrix is a column-permutation of the $n \times n$ lower triangular matrix, we give tight $\Theta(n \log n)$ bounds in both deterministic and randomized settings, and an $\Omega(n)$ quantum lower bound. \item For general half graphs, we observe that the problem is equivalent to a natural generalization of the famous nuts-and-bolts problem, leading to a tight $\Theta(n \log n)$ randomized bound. We also present a simple quicksort-style method that instantiates to a $O(n \log^2 n)$ randomized algorithm and a tight $O(n \log n)$ quantum algorithm. \end{itemize}

翻译：通过部分信息查询从已知图族中学习或重构未知图的问题，作为数学模型出现在多种情境中。访问图的最基本方式是通过\emph{边查询}，即算法可以单位成本查询其选定的一对顶点之间是否存在边。尽管在图重构背景下更强大的查询模型已得到广泛研究，但边查询这一基本模型似乎未受到同等关注。本文研究了在经典与量子设置下学习隐藏二分图（或其二分邻接矩阵）的边查询复杂度。我们重点关注学习匹配与半图：匹配的二分邻接矩阵是单位矩阵的行/列置换；半图的二分邻接矩阵是下三角矩阵（主对角线及以下元素全为1）的行/列置换。\begin{itemize} \item 对于规模为$n$的匹配，我们给出了紧确的确定性界$n(n-1)/2$与渐近紧确的随机化界$\Theta(n^2)$。van Apeldoorn等人近期工作[ICALP'21]已证明量子界为$\Theta(n^{1.5})$。\item 对于二分邻接矩阵为$n \times n$下三角矩阵列置换的半图，我们在确定性与随机化设置下均给出紧确的$\Theta(n \log n)$界，以及$\Omega(n)$的量子下界。\item 对于一般半图，我们观察到该问题等价于经典螺母螺栓问题的自然推广，从而得到紧确的$\Theta(n \log n)$随机化界。我们还提出一种快速排序风格的方法，可实例化为$O(n \log^2 n)$随机化算法与紧确的$O(n \log n)$量子算法。\end{itemize}