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Approximating Spanners and Directed Steiner Forest: Upper and Lower Bounds [article]

Eden Chlamtáč, Michael Dinitz, Guy Kortsarz, Bundit Laekhanukit
2016 arXiv   pre-print
In particular, our techniques allow us to give an O(n^3/5 + ϵ)-approximation for the Directed Steiner Forest problem (for arbitrary constant ϵ > 0) when all edges have uniform costs, improving the previous  ...  This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover.  ...  The current best approximation for Directed Steiner Forest isÕ(n 2/3 ) [7] , and also comes from intuition about spanners (directed spanners rather than distance preservers).  ... 
arXiv:1607.06068v1 fatcat:q5owvfqnrjcozmglkx3qb5tngy

Approximation algorithms for spanner problems and Directed Steiner Forest

Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, Grigory Yaroslavtsev
2013 Information and Computation  
We present an O( √ n log n)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices.  ...  A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph.  ...  Directed Steiner Forest Finally, we apply our techniques to the Directed Steiner Forest (DSF) problem, a fundamental network design problem on directed graphs.  ... 
doi:10.1016/j.ic.2012.10.007 fatcat:t37noy2zync67opj3qwuv5it4i

Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds [article]

Yun Kuen Cheung, Gramoz Goranci, Monika Henzinger
2016 arXiv   pre-print
We also build on the existing results on spanners, distance oracles and connected 0-extensions to show a number of upper bounds for general graphs, planar graphs, graphs that exclude a fixed minor and  ...  We introduce a novel black-box reduction to convert any lower bound on distortion for the SPR problem into a super-linear lower bound on the number of non-terminals, with the same distortion, for our problem  ...  Acknowledgements The authors thank Veronika Loitzenbauer and Harald Räcke for the helpful discussions. The  ... 
arXiv:1604.08342v1 fatcat:tckd4s34arcahn4y5ef7zmottq

Online Directed Spanners and Steiner Forests [article]

Elena Grigorescu, Young-San Lin, Kent Quanrud
2021 arXiv   pre-print
We present online algorithms for directed spanners and Steiner forests.  ...  We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion.  ...  Acknowledgements We thank the anonymous reviewers for comments and suggestions that helped improve the presentation.  ... 
arXiv:2103.04543v2 fatcat:r54ordqp7vc37f6kmsrhfvzjqe

Improved distributed steiner forest construction

Christoph Lenzen, Boaz Patt-Shamir
2014 Proceedings of the 2014 ACM symposium on Principles of distributed computing - PODC '14  
We also prove a matching lower bound ofΩ(k + min {s, √ n} + D) on the running time of any distributed approximation algorithm for the Steiner Forest problem.  ...  We present new distributed algorithms for constructing a Steiner Forest in the congest model.  ...  For Steiner Forest, the best approximation ratio known is 2 − 1/(t − k) [1] .  ... 
doi:10.1145/2611462.2611464 dblp:conf/podc/LenzenP14 fatcat:fi6l7yjw6zduveiwpcfdt3hp24

Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth

MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Dániel Marx
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus.  ...  As a first step, we show how to build a Steiner forest spanner for such graphs.  ...  Given any fixed ε > 0, a bounded genus graph G in (V in , E in ) and demand pairs D, we can compute in polynomial time a Steiner forest spanner H for G in with respect to demand set D.  ... 
doi:10.1145/1806689.1806720 dblp:conf/stoc/BateniHM10 fatcat:p5zoliylgvcgxdempnjvtap7gu

Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

Mohammadhossein Bateni, Mohammadtaghi Hajiaghayi, Dániel Marx
2011 Journal of the ACM  
We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus.  ...  As a first step, we show how to build a Steiner forest spanner for such graphs.  ...  Given any fixed ε > 0, a bounded genus graph G in (V in , E in ) and demand pairs D, we can compute in polynomial time a Steiner forest spanner H for G in with respect to demand set D.  ... 
doi:10.1145/2027216.2027219 fatcat:nwzh7vhoirg45pg2fewy35xpky

Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces [article]

Lee-Ad Gottlieb, Yair Bartal
2019 arXiv   pre-print
We give an algorithm that computes a (1+ϵ)-approximate Steiner forest in near-linear time n · 2^(1/ϵ)^O(ddim^2) ( n)^2.  ...  For Steiner tree our methods achieve an even better runtime n ( n)^(1/ϵ)^O(ddim^2) in doubling spaces.  ...  The Steiner forest problem inherits the same hardness of approximation of 96 95 from Steiner tree, and Agrawal et al. [AKR95] presented a 2-approximation for this problem. Borradaile et al.  ... 
arXiv:1904.03611v1 fatcat:rawiabjuyjh3pliyffpgfpanty

Graph Minors for Preserving Terminal Distances Approximately-Lower and Upper Bounds * †

Yun Cheung, Gramoz Goranci, Monika Henzinger
unpublished
We also build on the existing results on spanners, distance oracles and connected 0-extensions to show a number of upper bounds for general graphs, planar graphs, graphs that exclude a fixed minor and  ...  distances approximately.  ...  Discussion and Open Problems We note that there remain gaps between some of the best upper and lower bounds, e.g., for general graphs and distortion 3 − , the lower bound is Ω(k 6/5 ), while for distortion  ... 
fatcat:3dzzb6xnsjeg3kggt7gr4rya7q

Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners [article]

Omri Kahalon, Hung Le, Lazar Milenkovic, Shay Solomon
2022 arXiv   pre-print
To control the hop-length of paths, one can try to upper bound the spanner's hop-diameter, but naturally bounded hop-diameter spanners are more complex than spanners with unbounded hop-diameter, which  ...  While in the original metric, one can go from any point to any other via a direct edge, it is unclear how to efficiently navigate in the spanner: How can we translate the existence of a "good" path into  ...  to lower bounds by [AS87] and [LMS22] that apply to 1-spanners and (1+ǫ)-spanners for line metrics, respectively.  ... 
arXiv:2107.14221v2 fatcat:dwdkmbygvzggvm5zz372xbalhm

Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth [article]

MohammadHossein Bateni and MohammadTaghi Hajiaghayi and Dániel Marx
2009 arXiv   pre-print
We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus.  ...  As a first step, we show how to build a Steiner forest spanner for such graphs.  ...  To this end, we build a Steiner forest spanner for the input graph and the set of demands in Section 4.  ... 
arXiv:0911.5143v1 fatcat:wfrkxtitmrabtcdjhqco4meucm

Can't See the Forest for the Trees

Omri Kahalon, Hung Le, Lazar Milenković, Shay Solomon
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
Is it possible to efficiently navigate, on a sparse spanner, using 𝑘 hops and approximate distances, for 𝑘 close to 1 (say 𝑘 = 2)?  ...  Moreover, one can compute such spanners efficiently in the standard centralized and distributed settings.  ...  , and the United States National Science Foundation (NSF).  ... 
doi:10.1145/3519270.3538414 fatcat:oynmomltr5aqbdc56pi7fvfgla

Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights

Michael Dinitz, Guy Kortsarz, Zeev Nutov
2017 ACM Transactions on Algorithms  
node-costs c = {c v : v ∈ V }, and a lower profit bound , find a minimum node-cost subgraph of G of edge profit at least .  ...  We obtain ratio n 0.448 for Steiner k-Forest and Dial-a-Ride with unit weights, breaking the O( √ n) ratio barrier for this natural special case.  ...  node-costs c = {c v : v ∈ V }, and a profit lower bound .  ... 
doi:10.1145/3077581 fatcat:qlgfd6d2frcjrj2nqc2we5igia

A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting

MohammadHossein Bateni, Erik D. Demaine, MohammadTaghi Hajiaghayi, Dániel Marx
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which *  ...  We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size)  ...  Formally, we prove the existence of the following spanner extension step, which is just a straightforward application of Lemma 26.  ... 
doi:10.1145/2897518.2897549 dblp:conf/stoc/BateniDHM16 fatcat:gd2tnsws5rdyxc7orfzxzopxwy

Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees [article]

Markus Chimani, Joachim Spoerhase
2014 arXiv   pre-print
Finally, we show how to apply our results to obtain an (α+ε,O(n^0.5+ε))-approximation for light-weight directed α-spanners. For this, no non-trivial approximation algorithm has been known before.  ...  In a directed graph G with non-correlated edge lengths and costs, the network design problem with bounded distances asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs  ...  [2] Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and directed steiner forest.  ... 
arXiv:1409.6551v1 fatcat:hgvbyzv3j5hsjjvdnnxxludfrq
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