In a directed graph $D$ on vertex set $v_1,\dots ,v_n$, a \emph{forward arc} is an arc $v_iv_j$ where $i<j$. A pair $v_i,v_j$ is \emph{forward connected} if there is a directed path from $v_i$ to $v_j$ consisting of forward arcs. In the {\tt Forward Connected Pairs Problem} ({\tt FCPP}), the input is a strongly connected digraph $D$, and the output is the maximum number of forward connected pairs in some vertex enumeration of $D$. We show that {\tt FCPP} is in APX, as one can efficiently enumerate the vertices of $D$ in order to achieve a quadratic number of forward connected pairs. For this, we construct a linear size balanced bi-tree $T$ (an out-tree and an in-tree with same size which roots are identified). The existence of such a $T$ was left as an open problem motivated by the study of temporal paths in temporal networks. More precisely, $T$ can be constructed in quadratic time (in the number of vertices) and has size at least $n/3$. The algorithm involves a particular depth-first search tree (Left-DFS) of independent interest, and shows that every strongly connected directed graph has a balanced separator which is a circuit. Remarkably, in the request version {\tt RFCPP} of {\tt FCPP}, where the input is a strong digraph $D$ and a set of requests $R$ consisting of pairs $\{x_i,y_i\}$, there is no constant $c>0$ such that one can always find an enumeration realizing $c.|R|$ forward connected pairs $\{x_i,y_i\}$ (in either direction).

翻译：在顶点集为$v_1,\dots ,v_n$的有向图$D$中，一条\emph{前向弧}是指满足$i<j$的弧$v_iv_j$。若存在一条由前向弧组成的有向路径从$v_i$到$v_j$，则称对$v_i,v_j$是\emph{前向连通的}。在{\tt 前向连通对问题}（{\tt FCPP}）中，输入为强连通有向图$D$，输出为在$D$的某种顶点枚举中前向连通对的最大数量。我们证明{\tt FCPP}属于APX类，因为可以高效地枚举$D$的顶点以实现二次数量的前向连通对。为此，我们构造了一个线性大小的平衡双树$T$（一个出树和一个入树具有相同大小且根被识别）。此类$T$的存在性曾被提出作为开放问题，其动机源于对时间网络中时间路径的研究。更精确地说，$T$可在平方时间内（相对于顶点数）构造，且大小至少为$n/3$。该算法涉及一种具有独立意义的特定深度优先搜索树（Left-DFS），并表明每个强连通有向图都存在一个由回路构成的平衡分离器。值得注意的是，在{\tt FCPP}的请求版本{\tt RFCPP}中（输入为强有向图$D$及一组由对$\{x_i,y_i\}$构成的请求集$R$），不存在常数$c>0$使得总能找到一种枚举实现$c\cdot|R|$个前向连通对$\{x_i,y_i\}$（无论方向如何）。