article

Approximating k-spanner problems for k > 2

Authors:

David PelegAuthors Info & Claims

Theoretical Computer Science, Volume 337, Issue 1-3

Pages 249 - 277

https://doi.org/10.1016/j.tcs.2004.11.022

Published: 09 June 2005 Publication History

Abstract

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 d_H(u, w) ≤ kd_G(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(2^{log^1-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(n^2/(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.

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Censor-Hillel KDory MNewport CKeidar I(2018)Distributed Spanner ApproximationProceedings of the 2018 ACM Symposium on Principles of Distributed Computing10.1145/3212734.3212758(139-148)Online publication date: 23-Jul-2018
https://dl.acm.org/doi/10.1145/3212734.3212758
Dinitz MKortsarz GRaz R(2015)Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-SpannerACM Transactions on Algorithms10.1145/281837512:2(1-16)Online publication date: 31-Dec-2015
https://dl.acm.org/doi/10.1145/2818375
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Approximating k-spanner problems for k > 2

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Published: 09 June 2005

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Cited By

Censor-Hillel KDory MNewport CKeidar I(2018)Distributed Spanner ApproximationProceedings of the 2018 ACM Symposium on Principles of Distributed Computing10.1145/3212734.3212758(139-148)Online publication date: 23-Jul-2018
https://dl.acm.org/doi/10.1145/3212734.3212758
Dinitz MKortsarz GRaz R(2015)Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-SpannerACM Transactions on Algorithms10.1145/281837512:2(1-16)Online publication date: 31-Dec-2015
https://dl.acm.org/doi/10.1145/2818375
Barenboim LElkin MGavoille C(2015)A Fast Network-Decomposition Algorithm and Its Applications to Constant-Time Distributed ComputationPost-Proceedings of the 22nd International Colloquium on Structural Information and Communication Complexity - Volume 943910.1007/978-3-319-25258-2_15(209-223)Online publication date: 14-Jul-2015
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Purohit MPrakash BKang CZhang YSubrahmanian VMacskassy SPerlich CLeskovec JWang WGhani R(2014)Fast influence-based coarsening for large networksProceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining10.1145/2623330.2623701(1296-1305)Online publication date: 24-Aug-2014
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Dragan FAbu-Ata M(2014)Collective additive tree spanners of bounded tree-breadth graphs with generalizations and consequencesTheoretical Computer Science10.1016/j.tcs.2014.06.007547(1-17)Online publication date: 1-Aug-2014
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Dinitz MKortsarz GRaz R(2012)Label cover instances with large girth and the hardness of approximating basic k-spannerProceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I10.1007/978-3-642-31594-7_25(290-301)Online publication date: 9-Jul-2012
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Berman PBhattacharyya AMakarychev KRaskhodnikova SYaroslavtsev G(2011)Improved approximation for the directed spanner problemProceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I10.5555/2027127.2027129(1-12)Online publication date: 4-Jul-2011
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Dinitz MKrauthgamer RFortnow LVadhan S(2011)Directed spanners via flow-based linear programsProceedings of the forty-third annual ACM symposium on Theory of computing10.1145/1993636.1993680(323-332)Online publication date: 6-Jun-2011
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Dragan FFomin FGolovach P(2011)Approximation of minimum weight spanners for sparse graphsTheoretical Computer Science10.1016/j.tcs.2010.11.034412:8-10(846-852)Online publication date: 1-Mar-2011
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