Temporal Cliques Admit Sparse Spanners
Let G=(V,E) be an undirected graph on n vertices and lambda:E -> 2^{N} a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph {G}=(G,lambda) is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity - a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of Theta(n^2)-dense temporal graphs which admit no temporal spanner of density o(n^2). The natural question is then whether sparse temporal spanners always exist in some classes of dense graphs.
In this paper, we answer this question affirmatively, by showing that if the underlying graph G is a complete graph, then one can always find temporal spanners of density O(n log n). The best known result for complete graphs so far was that spanners of density binom{n}{2}- floor[n/4] = O(n^2) always exist. Our result is the first positive answer as to the existence of o(n^2) sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.
Dynamic networks
Temporal graphs
Temporal connectivity
Sparse spanners
Theory of computation~Sparsification and spanners
Theory of computation~Dynamic graph algorithms
134:1-134:14
Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management
This research was supported by ANR project ESTATE (ANR-16-CE25-0009-03) and NSERC of Canada.
A full version of this paper is available at https://arxiv.org/abs/1810.00104 [Casteigts et al., 2019].
We thank Cyril Gavoille for advice on the presentation of these results.
Arnaud
Casteigts
Arnaud Casteigts
LaBRI, Université de Bordeaux, CNRS, Bordeaux INP, France
https://orcid.org/0000-0002-7819-7013
Joseph G.
Peters
Joseph G. Peters
School of Computing Science, Simon Fraser University, Canada
https://orcid.org/0000-0002-2475-8145
Jason
Schoeters
Jason Schoeters
LaBRI, Université de Bordeaux, CNRS, Bordeaux INP, France
https://orcid.org/0000-0001-7257-5426
10.4230/LIPIcs.ICALP.2019.134
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Arnaud Casteigts, Joseph G. Peters, and Jason Schoeters
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