On Temporal Graph Exploration

TL;DR

This paper investigates the complexity and algorithms for exploring temporal graphs, including hardness results, efficient exploration strategies for specific graph classes, and online exploration methods under uncertainty.

Contribution

It provides new complexity bounds, explores efficient algorithms for certain graph classes, and introduces online exploration techniques for uncertain temporal graphs.

Findings

NP-hardness of approximating TEXP within certain ratios

Explicit constructions requiring quadratic exploration time

Efficient exploration algorithms for graphs with bounded treewidth, planarity, or grid structure

Abstract

A temporal graph is a graph in which the edge set can change from one time step to the next. 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 time step. For such temporal graphs with $n$ nodes, we show that it is \NP-hard to approximate TEXP with ratio $O(n^{1-\varepsilon})$ for every $\varepsilon>0$. We also provide an explicit construction of temporal graphs that require $\Theta(n^2)$ time steps to be explored. In the second part of the paper, we still consider temporal graphs that are connected in each time step, but we assume that the underlying graph (i.e. the graph that contains all edges that are present in the temporal graph in at least one time step) belongs to a specific class of graphs. Among other results, we show that temporal graphs can be explored in $O(n^{1.5}k^{1.5}\log n)$ time steps if the underlying graph has treewidth $k$, in $O(n^{1.8}\log n)$ time steps if the underlying graph is planar, and in $O(n\log^3 n)$ time steps if the underlying graph is a $2\times n$ grid. In the third part of the paper, we consider settings where the graphs in future time steps are not known and the exploration schedule is constructed online. We replace the connectedness assumption by a weaker assumption and show that $m$-edge temporal graphs with regularly present edges and with probabilistically present edges can be explored online in $O(m)$ time steps and $O(m \log n)$ time steps with high probability, respectively. We finally show that the latter result can be used to obtain a distributed algorithm for the gossiping problem in random temporal graphs.