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
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
(1 + &egr;&Bgr;)-spanner constructions for general graphs
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Generating Sparse Spanners for Weighted Graphs
SWAT '90 Proceedings of the 2nd Scandinavian Workshop on Algorithm Theory
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
Extremal Graph Theory
A PTAS for the Sparsest Spanners Problem on Apex-Minor-Free Graphs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Dynamic and Efficient Key Management for Access Hierarchies
ACM Transactions on Information and System Security (TISSEC)
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Additive spanners and (α, β)-spanners
ACM Transactions on Algorithms (TALG)
Approximation of minimum weight spanners for sparse graphs
Theoretical Computer Science
Directed spanners via flow-based linear programs
Proceedings of the forty-third annual ACM symposium on Theory of computing
Journal of Computer and System Sciences
Improved approximation for the directed spanner problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Label cover instances with large girth and the hardness of approximating basic k-spanner
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Approximation algorithms for spanner problems and Directed Steiner Forest
Information and Computation
Hi-index | 5.23 |
Given a graph G = (V, E), a subgraph G' = (V, H), H ⊆ E is a k-spanner of G if for any pair of vertices u, w ∈ V it satisfies dH(u, w) ≤ kdG(u, w). The basic k-spanner problem is to find a k-spanner of a given graph G with the smallest possible number of edges. This paper considers approximation algorithms for this and some related problems for k 2, known to be Ω(2log1-µn)-inapproximable. 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 related 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 Õ(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 reduces the studied algorithmic approximability questions on k-spanners to purely graph-theoretical questions concerning the existence of certain combinatorial objects in families of graphs.