Non-strict Temporal Exploration

A temporal graph G = < G(1),..., G(L) > is a sequence of graphs G(i) subset of 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 G admits a sequence W of consecutively crossed edges e is an element of G, such that W visits all vertices at least once and that each e is an element of W is crossed at a timestep t' is an element of [L] such that t' >= t, where t is the timestep during which the previous edge was crossed. This variant of the problem is shown to be 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(1/2) (-) (epsilon))-inapproximability and O(n(1-epsilon))-inapproximability in the c = 2 and c >= 3 cases, respectively. For graphs with temporal diameter c = 2, we also prove an O(root n log n) upper bound on worst-case time required for exploration, as well as an Omega(root n) lower bound.