In a temporal graph the edge set dynamically changes over time according to a
set of time-labels associated with each edge that indicates at which time-steps
the edge is available. Two vertices are connected if there is a path connecting
them in which the edges are traversed in increasing order of their labels. Nous
study the problem of scheduling the availability time of the edges of a
temporal graph in such a way that all pairs of vertices are connected within a
given maximum allowed time $a$ and the overall number of labels is minimized.
The problem, known as \emph{Minimum Aged Labeling} (MAL), has several
applications in logistics, distribution scheduling, and information spreading
in social networks, where carefully choosing the time-labels can significantly
reduce infrastructure costs, fuel consumption, or greenhouse gases.
The problem MAL has previously been proved to be NP-complete on undirected
graphs and \APX-hard on directed graphs. Dans ce document, we extend our knowledge
on the complexity and approximability of MAL in several directions. We first
show that the problem cannot be approximated within a factor better than
$Ô(\log n)$ when $a\geq 2$, unless $\text{P} = \text{NP}$, and a factor better
than $2^{\log ^{1-\epsilon} n}$ when $a\geq 3$, unless $\text{NP}\subseteq
\text{DTIME}(2^{\text{polylog}(n)})$, where $n$ is the number of vertices in
the graph. Then we give a set of approximation algorithms that, under some
conditions, almost match these lower bounds. In particular, we show that the
approximation depends on a relation between $a$ and the diameter of the input
graph.
We further establish a connection with a foundational optimization problem on
static graphs called \emph{Diameter Constrained Spanning Subgraph} (DCSS) et
show that our hardness results also apply to DCSS.

Cet article explore les excursions dans le temps et leurs implications.

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