There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
An optimal synchronizer for the hypercube
SIAM Journal on Computing
Generating sparse spanners for weighted graphs
SWAT '90 Proceedings of the second Scandinavian workshop on Algorithm theory
New sparseness results on graph spanners
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Journal of Algorithms
NP-completeness of minimum spanner problems
Discrete Applied Mathematics
SIAM Journal on Discrete Mathematics
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
On the Hardness of Approximation Spanners
APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
All pairs almost shortest paths
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
The Client-Server 2-Spanner Problem and Applications to Network Design
The Client-Server 2-Spanner Problem and Applications to Network Design
Delaunay graphs are almost as good as complete graphs
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Exact and Approximate Distances in Graphs - A Survey
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
Sparsification of influence networks
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining
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Given a graph G = (V, E), a subgraph G′ = (V, H), H ⊆ E is a k-spanner (respectively, k-DSS) of G if for any pair of vertices u, w ∈ V it satisfies dH(u, w) ≤ k ċ dG(u, w) (resp., dH(u, w) ≤ k). The basic k- spanner (resp., k-DSS ) problem is to find a k-spanner (resp., k-DSS) of a given graph G with the smallest possible number of edges. This paper considers approximation algorithms for these and some related problems for k 2. Both problems are known to be Ω(2log1-Ɛ n)- inapproximable [11,13]. The basic k-spanner problem over undirected graphs with k 2 has been given a sublinear ratio approximation algorithm (with ratio roughly O(n2/k+1)), but no such algorithms were known for other variants of the problem, including the directed and the client-server variants, as well as for the k-DSS problem. We present the first approximation algorithms for these problems with sublinear approximation ratio. The second contribution of this paper is in characterizing some wide families of graphs on which the problems do admit a logarithmic and a polylogarithmic approximation ratios. These families are characterized as containing graphs that have optimal or "near-optimal" spanners with certain desirable properties, such as being a tree, having low arboricity or having low girth. All our results generalize to the directed and the client-server variants of the problems. As a simple corollary, we present an algorithm that given a graph G builds a subgraph with O(n) edges and stretch bounded by the tree-stretch of G, namely the minimum maximal stretch of a spanning tree for G. The analysis of our algorithms involves the novel notion of edge-dominating systems developed in the paper. The technique introduced in the paper enables us to reduce the studied algorithmic questions of approximability of the k-spanner and k-DSS problems to purely graph-theoretical questions concerning the existence of certain combinatorial objects in families of graphs.