On Temporal Graph Exploration

Abstract

A temporal graph is a graph in which the edge set can change from step to step. The temporal graph exploration problem TEXP is the problem of computing a foremost exploration schedule for a temporal graph, i.e., a temporal walk that starts at a given start node, visits all nodes of the graph, and has the smallest arrival time. In the first part of the paper, we consider only undirected temporal graphs that are connected at each step. For such temporal graphs with n nodes, we show that it is NP-hard to approximate TEXP with ratio O(n 1−ε ) for any ε > 0. We also provide an explicit construction of temporal graphs that require Θ(n 2 ) steps to be explored. We then consider TEXP under the assumption that the underlying graph (i.e. the graph that contains all edges that are present in the temporal graph in at least one step) belongs to a specific class of graphs. Among other results, we show that temporal graphs can be explored in O(n 1.5 k 2 log n) steps if the underlying graph has treewidth k and in O(n log 3 n) steps if the underlying graph is a 2 × n grid. In the second part of the paper, we replace the connectedness assumption by a weaker assumption and show that m-edge temporal graphs with regularly present edges and with random edges can always be explored in O(m) steps and O(m log n) steps with high probability, respectively. We finally show that the latter result can be used to obtain a distributed algorithm for the gossiping problem.

An Introduction to Temporal Graphs: An Algorithmic Perspective*

“…The above inapproximability result has been recently improved [23] to O(n 1−ε ) for any ε > 0. They first present a family of continuously connected temporal graphs that require (n 2 ) steps to be explored.…”

“…Assume additionally that the resulting temporal graph has connected instances. It was proved in [23] that such a temporal graph can be explored in O(n) steps. The idea is to first round down each R e to the nearest power of 2 and obtain a new set of values R e .…”

“…In [23], the authors additionally studied the Temporal Exploration problem in other interesting restricted families of continuously connected temporal graphs, such as those whose underlying graph has treewidth k (a work explicitly concerned with the treewidth of temporal graphs and its relation to the treewidth of static graphs is [47]), is a 2 × n grid, a cycle, a cycle with a chord, or a bounded-degree planar graph, for which they provided upper bounds on exploration time. Several of these results were proved in the following interesting way.…”

“…This will imply that the total cost of T has been distributed to the edges of G in such a way that every edge of G has received a charge upper bounded by a constant H and as G has O(n) edges, the total cost of T must have been at most H · O(n) = O(n), which is what we wanted to prove. In fact, H turns out to be equal to 4c, where c is the constant assumed in the R e /c minimum consecutive unavailability of an edge e. For the precise charging mechanism, which is quite technical, relying on the facts that every intercomponent edge must have cost at least 2 k (otherwise it could be used as a swapping edge to obtain a tree cheaper than T ) and that in every step at least one of the edges of each E i must be present (due to the continuous connectivity assumption), the reader is referred to [23].…”

Sliding Window Temporal Graph Coloring

“…Now the classical computational problems have to be appropriately redefined in the temporal setting in order to properly capture the notion of time. Motivated by the fact that, due to causality, information in temporal graphs can "flow" only along sequences of edges whose time-labels are increasing, most temporal graph parameters and optimization problems that have been studied so far are based on the notion of temporal paths and other "path-related" notions, such as temporal analogues of distance, reachability, separators, diameter, exploration, and centrality [2,3,14,15,18,28,32,40].…”