Let @math be a labeled graph on @math vertices with @math a locally injective mapping that assigns to every edge a single integer label. The label is seen as a discrete time when the edge is present. This graph is temporally connected if a path exists with increasing labels from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (JCSS 2002) asked whether, given such a labeled graph, a sparse subset of edges can always be found that preserves temporal connectivity if the other edges are removed -- we call such subsets temporal spanners . Recently, Axiotis and Fotakis (ICALP 2016) answered negatively, exhibiting a family of minimally connected temporal graphs with @math edges. The natural question then becomes whether sparse spanners can be found in specific classes of dense graphs. In this article, we settle the question positively for complete graphs, showing that one can always remove all but @math edges, whatever the labels, while preserving temporal connectivity. The best approach so far led to removing only @math edges, leaving the asymptotic density of the graph unchanged (, ToCS 2017). We start by observing that the same argument can be generalized to removing @math edges (a sixth of the edges). Then, using a completely different approach, we establish a gradual set of results, showing that a quarter of the edges can be removed, then half of the edges, and eventually all but @math edges. This result is robust in the sense that it extends, under mild assumptions, to more general models of temporal cliques where the labels may not be locally unique and a same edge may have several labels. The main open question is now to understand where the separation occurs between graphs that admit sparse spanners and graphs that do not.
In this paper, we refer to subsets of edges that preserve temporal connectivity as temporal spanners , inspired by their analogues in standard graphs and geometry (see, for example, @cite_0 , @cite_3 , and @cite_15 ), and to the process of removing edges while preserving temporal connectivity as simplification (or sparsification). The model considered in @cite_17 assumes that every edge in @math is given a single presence time, which allows one to formulate the problem in terms of subsets of edges . In fact, it is immediately shown in @cite_17 that one may label an hypercube so that none of its edges can be removed, leading to an update of the question to finding a subset of density @math .
@cite_0 @cite_15 @cite_3 @cite_17
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Part I. Introduction: 1. Introduction 2. Algorithms and graphs 3. The algebraic computation-tree model Part II. Spanners Based on Simplical Cones: 4. Spanners based on the Q-graph 5. Cones in higher dimensional space and Q-graphs 6. Geometric analysis: the gap property 7. The gap-greedy algorithm 8. Enumerating distances using spanners of bounded degree Part III. The Well Separated Pair Decomposition and its Applications: 9. The well-separated pair decomposition 10. Applications of well-separated pairs 11. The Dumbbell theorem 12. Shortcutting trees and spanners with low spanner diameter 13. Approximating the stretch factor of Euclidean graphs Part IV. The Path Greedy Algorithm: 14. Geometric analysis: the leapfrog property 15. The path-greedy algorithm Part V. Further Results and Applications: 16. The distance range hierarchy 17. Approximating shortest paths in spanners 18. Fault-tolerant spanners 19. Designing approximation algorithms with spanners 20. Further results and open problems. Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straight-line distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less than √10 times |ab|, where |ab| is the straight-line Euclidean distance between a and b. Thus, a shortest path in this planar graph is less than about 3 times longer than the corresponding straight-line distance. The triangulation that has this property is the L 1 metric Delaunay triangulation for the set 5. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O(n log n) time. Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints "communicated." We will call such a graph a temporal network. To model the notion that information in such a network "flows" only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. We provide results on two types of problems for temporal networks. First, we consider connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes. The natural analogue of Menger's Theorem for node-disjoint paths fails in general for time-respecting paths; we give a non-trivial characterization of those graphs for which the theorem does hold in terms of an excluded subdivision theorem, and provide a polynomial-time algorithm for connectivity on this class of graphs. (The problem on general graphs is NP-complete.) We then define and study the class of inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow.