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 vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex to vertex is a path from to where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (, )-journey for any pair of vertices , , ≠ . 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 availability instances for all edges and aims for temporal connectivity with very small ; 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 . 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 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 = . 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 , i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least log labels.