Better late, then? The hardness of choosing delays to meet passenger demands in temporal graphs

TL;DR

This work introduces DelayBetter and its variants for adjusting edge times in temporal graphs to satisfy passenger-arrival demands, linking Delay Management with temporal graph reachability. It shows Path-DB and δ-Path-DB are solvable in polynomial time via a linear-programming formulation, with corollaries that DB and δ-DB are tractable on trees, and it provides a fixed-parameter tractable algorithm parameterized by the number of demands $|D|$ and the size $ ho$ of a minimum feedback-edge set. The paper also establishes strong hardness results: ($1$-)DB is NP-hard on bounded-degree graphs with $T_{ ext{max}}=2$, and planar instances with $T_{ ext{max}}=19$ (for δ up to 10) remain hard, delineating clear tractability boundaries. Overall, the results connect delay-management concepts to temporal-graph modification problems and offer practical and theoretical insights into when targeted passenger demands can be feasibly satisfied.

Abstract

In train networks, carefully-chosen delays may be beneficial for certain passengers, who would otherwise miss some connection. Given a simple (directed or undirected) temporal graph and a set of passengers (each specifying a starting vertex, an ending vertex, and a desired arrival time), we ask whether it is possible to delay some of the edges of the temporal graph to realize all the passengers' demands. We call this problem DelayBetter (DB), and study it along with two variants: in $δ$-DelayBetter, each delay must be of at most $δ$; in ($δ$-)Path DB, passengers also fully specify the vertices they should visit on their journey. On the positive side, we give a polynomial-time algorithm for Path DB and $δ$-Path DB, and obtain as a corollary a polynomial-time algorithm for DB and $δ$-DB on trees. We also provide an fpt algorithm for both problems parameterized by the size of the graph's Feedback Edge Set together with the number of passengers. On the negative side, we show NP-completeness of ($1$-)DB on bounded-degree temporal graphs even when the lifetime is $2$, and of ($10$-)DB on bounded-degree planar temporal graphs of lifetime $19$. Our results complement previous work studying reachability problems in temporal graphs with delaying operations. This is to our knowledge the first such problem in which the aim is to facilitate travel between specific points (as opposed to facilitating or impeding a broadcast from one or many sources).

Figures (1)

• Figure 1: A temporal graph on 6 vertices. Consider the case where passengers at each of $u,v$, and $w$ wish to travel to each of $x,y$, and $z$ respectively, arriving at or before time 4. Then delaying the edge from $w$ to $x$ by at least $2$ is necessary for the two leftmost passengers to arrive on time, but entails that the passenger starting at $w$ cannot arrive at $z$ before time $5$.

Theorems & Definitions (12)

• Definition 1: Temporal graph, temporal path
• Definition 2: Delaying
• Lemma 3
• Lemma 4: $\ast$
• Lemma 4: $\ast$
• Lemma 5
• Theorem 6
• Corollary 7
• Theorem 8
• Theorem 9