The complexity of optimal design of temporally connected graphs

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of $n$ vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex $u$ to vertex $v$ is a path from $u$ to $v$ where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a $(u,v)$-journey for any pair of vertices $u,v,~u\not= v$. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can \emph{freely choose} availability instances for all edges and aims for temporal connectivity with very small \emph{cost}; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in $n$. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead \emph{given} a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless $P=NP$. On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be \emph{minimal}, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least $n \log{n}$ labels.