Traveling salesman problems in temporal graphs

An Introduction to Temporal Graphs: An Algorithmic Perspective*

“…Maximum Matching can be solved in polynomial time by the famous Edmonds' algorithm [22] (the time is O( √ |V | · |E|) by the algorithm of [51]). Now consider the following temporal version of the problem, called Temporal Matching in [60]. In this problem, we are given a temporal graph D = (V , A) and we are asked to decide whether there is a maximum matching M of the underlying static graph of D that can be made temporal by selecting a single label l ∈ λ(e) for every edge e ∈ M. For a single-labeled matching to be temporal, it suffices to guarantee that no two of its edges have the same label.…”

“…In this problem, we are given a temporal graph D = (V , A) and we are asked to decide whether there is a maximum matching M of the underlying static graph of D that can be made temporal by selecting a single label l ∈ λ(e) for every edge e ∈ M. For a single-labeled matching to be temporal, it suffices to guarantee that no two of its edges have the same label. Temporal Matching was proved in [60] to be NP-complete. Then, the problem of computing a maximum cardinality temporal matching is immediately NP-hard, because if we could compute such a maximum temporal matching in polynomial time, we could then compare its cardinality to the cardinality of a maximum static matching and decide Temporal Matching in polynomial time.…”

“…NP-completeness of Temporal Matching can be proved by the sequence of polynomial-time reductions: Balanced 3SAT ≤ P Balanced Union Labeled Matching ≤ P Temporal Matching. In Balanced 3SAT, which is known to be NP-complete, every variable x i appears n i times negated and n i times non-negated and in Balanced Union Labeled Matching we are given a bipartite graph G = ((X, Y ), E), labels L = {1, 2, ..., h}, and a labeling λ : E → 2 L , every node u i ∈ X has precisely two neighbors v ij ∈ Y , and, additionally, both edges of u i have the same number of labels, and we must decide whether there is a maximum matching M of G such that e∈M λ(e) = L [60].…”

“…Another interesting problem is the Temporal Exploration problem [60]. In this problem, we are given a temporal graph and the goal is to visit all nodes of the temporal graph by a temporal walk that possibly revisits nodes, minimizing the arrival time.…”

“…In contrast to these, it was proved in [60] that there exists some constant c > 0 such that Temporal Exploration cannot be approximated within cn unless P = NP, even in temporal graphs consisting, in every instance, of two strongly connected components (weakly connected with each other), by presenting a gap introducing reduction from Hampath. Additionally, it was proved that even the special case in which every instance of the temporal graph is strongly connected, cannot be approximated within (2 − ε), for every constant ε > 0, unless P = NP.…”