Temporalizing digraphs via linear-size balanced bi-trees

TL;DR

This work shows that for strongly connected digraphs, a linear-size balanced bi-tree consisting of an in-tree and an out-tree with a common root can be constructed in $O(n^2)$ time and has size at least $n/3$, providing a concrete route to a $1/18$-approximation for the Forward Connected Pairs Problem (FCPP). The approach hinges on left-maximal DFS trees and a cyclic balanced separator, yielding structural decompositions that underpin the bi-tree existence and enabling implications for temporal path scheduling (MRET) and related problems. The paper also introduces bi-labels and transfer techniques to quantify and realize bi-trees, and investigates the request-based variant RFCPP, showing fundamental limits (no constant-factor guarantee in general) and proposing an $O( obreak ext{log }n)$ forward-cover conjecture. Collectively, these results connect graph scheduling, temporal networks, and combinatorial decompositions, advancing our understanding of how to efficiently realize large fractions of forward-connected relationships in digraphs. Key contributions include the $n/3$-size linear-time construct, the left-maximal DFS methodology, the cyclic balanced separator, and the explicit positive/negative results for FCPP and RFCPP, with implications for APX-hardness reductions and potential extensions to weighted and cover-based variants.

Abstract

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).

Figures (2)

• Figure 1: The digraph $D$ in the proof of Proposition \ref{['prop:example-sec2']}. An arc between a block and a particular vertex stands for all the arcs between each vertex of the block and the particular vertex. The arc $yx$ is not drawn.
• Figure 2: Left: a strong digraph (all arcs) with a left-maximal dfs-tree $T$ (plain arcs) and the $(I,C,O)$ decomposition associated to $T_{x,y}$. Nodes are numbered according to the corresponding dfs traversal. Right: the cycle spanning $C$ (plain arcs), two in-trees spanning $I$ and one out-tree spanning $O$.

Theorems & Definitions (11)

• Theorem 1
• Proposition 1
• Theorem 2
• Proposition 2
• Proposition 3
• Corollary 1
• Theorem 3
• Theorem 4
• Corollary 2
• Proposition 4