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

TL;DR

This paper investigates the computational complexity of delaying edges in temporal graphs to meet specific passenger travel demands, providing polynomial algorithms for some cases and NP-completeness results for others.

Contribution

It introduces the DelayBetter problem, offers polynomial solutions for certain variants, and proves NP-completeness for others, advancing understanding of delay optimization in temporal graphs.

Findings

Polynomial algorithms for Path DB and $oldsymbol{ extdelta}$-Path DB.

NP-completeness of DB on bounded-degree graphs with short lifetime.

FPT algorithm parameterized by feedback edge set size and passenger count.

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 $\delta$-DelayBetter, each delay must be of at most $\delta$; in ($\delta$-)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 $\delta$-Path DB, and obtain as a corollary a polynomial-time algorithm for DB and $\delta$-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).