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

2016
*
arXiv
*
pre-print

In particular, our techniques allow us to give an O(n^3/5 + ϵ)-

arXiv:1607.06068v1
fatcat:q5owvfqnrjcozmglkx3qb5tngy
*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). ...##
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Approximation algorithms for spanner problems and Directed Steiner Forest

2013
*
Information and Computation
*

We present an O( √ n log n)-

doi:10.1016/j.ic.2012.10.007
fatcat:t37noy2zync67opj3qwuv5it4i
*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. ...##
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Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds
[article]

2016
*
arXiv
*
pre-print

We also build on the existing results on

arXiv:1604.08342v1
fatcat:tckd4s34arcahn4y5ef7zmottq
*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 ...##
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Online Directed Spanners and Steiner Forests
[article]

2021
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arXiv
*
pre-print

We present online algorithms for

arXiv:2103.04543v2
fatcat:r54ordqp7vc37f6kmsrhfvzjqe
*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. ...##
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Improved distributed steiner forest construction

2014
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Proceedings of the 2014 ACM symposium on Principles of distributed computing - PODC '14
*

We also prove a matching

doi:10.1145/2611462.2611464
dblp:conf/podc/LenzenP14
fatcat:fi6l7yjw6zduveiwpcfdt3hp24
*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] . ...##
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Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth

2010
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Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10
*

We give the first polynomial-time

doi:10.1145/1806689.1806720
dblp:conf/stoc/BateniHM10
fatcat:p5zoliylgvcgxdempnjvtap7gu
*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. ...##
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Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

2011
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Journal of the ACM
*

We give the first polynomial-time

doi:10.1145/2027216.2027219
fatcat:nwzh7vhoirg45pg2fewy35xpky
*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. ...##
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Near-linear time approximation schemes for Steiner tree and forest in low-dimensional spaces
[article]

2019
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arXiv
*
pre-print

We give an algorithm that computes a (1+ϵ)-

arXiv:1904.03611v1
fatcat:rawiabjuyjh3pliyffpgfpanty
*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. ...##
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Graph Minors for Preserving Terminal Distances Approximately-Lower and Upper Bounds * †

unpublished

We also build on the existing results on

fatcat:3dzzb6xnsjeg3kggt7gr4rya7q
*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 ...##
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Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners
[article]

2022
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arXiv
*
pre-print

To control the hop-length of paths, one can try to

arXiv:2107.14221v2
fatcat:dwdkmbygvzggvm5zz372xbalhm
*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. ...##
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Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth
[article]

2009
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arXiv
*
pre-print

We give the first polynomial-time

arXiv:0911.5143v1
fatcat:wfrkxtitmrabtcdjhqco4meucm
*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. ...##
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Can't See the Forest for the Trees

2022
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Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing
*

Is it possible to efficiently navigate, on a sparse

doi:10.1145/3519270.3538414
fatcat:oynmomltr5aqbdc56pi7fvfgla
*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). ...##
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Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights

2017
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ACM Transactions on Algorithms
*

node-costs c = {c v : v ∈ V },

doi:10.1145/3077581
fatcat:qlgfd6d2frcjrj2nqc2we5igia
*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*. ...##
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A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting

2016
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Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016
*

Our second major contribution required for the planar group

doi:10.1145/2897518.2897549
dblp:conf/stoc/BateniDHM16
fatcat:gd2tnsws5rdyxc7orfzxzopxwy
*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. ...##
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Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees
[article]

2014
*
arXiv
*
pre-print

Finally, we show how to apply our results to obtain an (α+ε,O(n^0.5+ε))-

arXiv:1409.6551v1
fatcat:hgvbyzv3j5hsjjvdnnxxludfrq
*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*. ...
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