Non-strict Temporal Exploration

A temporal graph \(\mathcal {G} = \langle G_1, ..., G_L\rangle \) is a sequence of graphs \(G_i \subseteq G\), for some given underlying graph G of order n. We consider the non-strict variant of the Temporal Exploration problem, in which we are asked to decide if \(\mathcal {G}\) admits a sequence W of consecutively crossed edges \(e \in G\), such that W visits all vertices at least once and that each \(e \in W\) is crossed at a timestep \(t' \in [L]\) such that \(t' \ge t\), where t is the timestep during which the previous edge was crossed. This variant of the problem is shown to be \(\textsf {NP}\)-complete. We also consider the hardness of approximating the exploration time for yes-instances in which our order-n input graph satisfies certain assumptions that ensure exploration schedules always exist. The first is that each pair of vertices are contained in the same component at least once in every period of n steps, whilst the second is that the temporal diameter of our input graph is bounded by a constant c. For the latter of these two assumptions we show \(O(n^{\frac{1}{2}-\varepsilon })\)-inapproximability and \(O(n^{1-\varepsilon })\)-inapproximability in the \(c=2\) and \(c \ge 3\) cases, respectively. For graphs with temporal diameter \(c=2\), we also prove an \(O(\sqrt{n} \log n)\) upper bound on worst-case time required for exploration, as well as an \(\varOmega (\sqrt{n})\) lower bound.