Approximating spanners and directed steiner forest: Upper and lower bounds

Eden Chlamtáč, Michael Dinitz, Guy Kortsarz, Bundit Laekhanukitx

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

18 Scopus citations

Abstract

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA 16, Bodwin-Williams SODA 16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O(n3=5+)- Approximation for distance preservers and pairwise spanners (for arbitrary constant ϵ > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O(log n)-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O(1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O(n3=5+)- Approximation for the Directed Steiner Forest problem (for arbitrary constant > 0) when all edges have uniform costs, improving the previous best O(n2=3+ϵ)- Approximation due to Berman et al. [ICALP '11] (which holds for general edge costs).

Original languageEnglish
Title of host publication28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
EditorsPhilip N. Klein
PublisherAssociation for Computing Machinery
Pages534-553
Number of pages20
ISBN (Electronic)9781611974782
DOIs
StatePublished - 1 Jan 2017
Event28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain
Duration: 16 Jan 201719 Jan 2017

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume0

Conference

Conference28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
Country/TerritorySpain
CityBarcelona
Period16/01/1719/01/17

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