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Tree Spanners for Subgraphs and Related Tree Covering Problems
[chapter]

2000
*
Lecture Notes in Computer Science
*

*For*the case k = 2, we develop a model in terms of an equivalent

*tree*

*covering*

*problem*,

*and*use this to show N P-hardness. ... We conclude by discussing some

*problems*

*related*to the model

*for*the case k = 2. ...

*Related*

*tree*

*covering*

*problems*Motivated by the subtree

*covering*

*problem*that has appeared in the context of Steiner

*Tree*2-

*Spanner*, we now examine different variations of this kind of

*covering*

*problems*...

##
###
Shortest-path metric approximation for random subgraphs

2006
*
Random structures & algorithms (Print)
*

With the goal of speeding up the repeated queries

doi:10.1002/rsa.20150
fatcat:hu4fvapi3vfddbp7pfcwsersgu
*and*saving space, we describe the construction of a sparse*subgraph*Q ⊂ G which contains an approximately optimal solution*for*any such*problem*on a random ... The number of edges in Q is O(p −c n 1+2/c log n)*for*edge-induced random*subgraphs**and*O(p −2c n 1+2/c log 2 n)*for*vertexinduced random*subgraphs*, where n is the number of vertices in G, p the sampling ...*Relation*to*spanners*. Our notion of metric approximation*for*random subraphs is reminiscent of the notion of a*spanner*, a sparse*subgraph*approximating distances in the original graph. ...##
###
Page 10292 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

“This paper considers approximation algorithms

*for*these*and*some*related**problems**for*k > 2. Both*problems*are known to be Q(2'°2 °")-inapproximable. ... “In this paper we present a 2-approximation NC (*and*RNC) algorithm*for*connected vertex*cover*(*and**tree**cover*). The NC algorithm runs in O(log? n) time using O(A? ...##
###
connected Joins in Graphs
[chapter]

2001
*
Lecture Notes in Computer Science
*

*problem*;

*subgraph*isomorphism. ... We study the

*problem*of finding a connected join

*covering*a given subset of vertices of the graph, that is a Steiner

*tree*which is a join at the same time. ... Note that a

*tree*λ-

*spanner*is a connected λ-

*tree*

*covering*S = V (G), a special case of our

*problem*. ...

##
###
Independent tree spanners: fault-tolerant spanning trees with constant distance guarantees

2001
*
Discrete Applied Mathematics
*

*For*t ¡ 3, we give a (constructive) linear-time algorithm to decide whether a pair of independent

*tree*t-

*spanners*exist. We also show that the

*problem*

*for*arbitrary t¿4 in NP-complete. ...

*For*any ÿxed rational parameter t¿1, a (

*tree*) t-

*spanner*of a graph G is a spanning

*subgraph*(

*tree*) T in G such that the distance between every pair of vertices in T is at most t times their distance in ... Acknowledgements We thank the anonymous referees

*for*their helpful comments. ...

##
###
Page 4405 of Mathematical Reviews Vol. , Issue 2002F
[page]

2002
*
Mathematical Reviews
*

{

*For*the entire collection see MR 2001c:68012.} 2002f:68136 68R10 05C85 68Q17 Handke, Dagmar (D-K NST; Konstanz); Kortsarz, Guy (IL-OPEN; Tel Aviv)*Tree**spanners**for**subgraphs**and**related**tree**covering*...*For*the case k = 2, we develop a model in terms of an equivalent*tree**covering**problem*,*and*use this to show NP- hardness. ...##
###
Approximating k-Spanner Problems for k > 2
[chapter]

2001
*
Lecture Notes in Computer Science
*

of the

doi:10.1007/3-540-45535-3_8
fatcat:q32xzzhq4bdonjkjmpu6gmjvla
*problem*, including the directed*and*the client-server variants, as well as*for*the*related*k-DSS*problem*. ... This paper considers approximation algorithms*for*this*and*some*related**problems**for*k > 2, known to be (2 log 1− n )inapproximable. ... In particular,*for*any graph G this algorithm builds a*subgraph*H of G with Õ(n) edges*and*stretch(H ) S*TREE*(G). ...##
###
Approximating k-spanner problems for k>2

2005
*
Theoretical Computer Science
*

of the

doi:10.1016/j.tcs.2004.11.022
fatcat:pw5dbl6iyzbtnghfk4va7mtxnq
*problem*, including the directed*and*the client-server variants, as well as*for*the*related*k-DSS*problem*. ... This paper considers approximation algorithms*for*this*and*some*related**problems**for*k > 2, known to be (2 log 1− n )inapproximable. ... In particular,*for*any graph G this algorithm builds a*subgraph*H of G with Õ(n) edges*and*stretch(H ) S*TREE*(G). ...##
###
Spanners in sparse graphs

2011
*
Journal of computer and system sciences (Print)
*

In particular,

doi:10.1016/j.jcss.2010.10.002
fatcat:4sss5iadljgvfbqmf23ge4grme
*for*every t 4, the*problem*of finding a*tree*t-*spanner*is NP-complete on K 6 -minor-free graphs. ... In 1998, Fekete*and*Kremer showed that on unweighted planar graphs deciding whether G admits a*tree*t-*spanner*is polynomial time solvable*for*t 3*and*is NP-complete when t is part of the input. ...*Related*work. Substantial work has been done on the*Tree*t-*spanner**problem*, also known as the minimum stretch spanning*tree**problem*. ...##
###
Spanners in Sparse Graphs
[chapter]

2008
*
Lecture Notes in Computer Science
*

In particular,

doi:10.1007/978-3-540-70575-8_49
fatcat:ul4bhmcaw5fihek7cgdzshxz7u
*for*every t 4, the*problem*of finding a*tree*t-*spanner*is NP-complete on K 6 -minor-free graphs. ... In 1998, Fekete*and*Kremer showed that on unweighted planar graphs deciding whether G admits a*tree*t-*spanner*is polynomial time solvable*for*t 3*and*is NP-complete when t is part of the input. ...*Related*work. Substantial work has been done on the*Tree*t-*spanner**problem*, also known as the minimum stretch spanning*tree**problem*. ...##
###
Tree Spanners

1995
*
SIAM Journal on Discrete Mathematics
*

Furthermore, linear-time algorithms are derived

doi:10.1137/s0895480192237403
fatcat:wxag2mbkvnbzbbdvtkidhdm2gi
*for*verifying*tree**spanners**and*quasi-*tree**spanners*. ... The results*for**tree**spanners*on undirected graphs are extended to "quasi-*tree**spanners*" on digraphs. ... The authors thank the referees*for*useful suggestions. ...##
###
Online Directed Spanners and Steiner Forests
[article]

2021
*
arXiv
*
pre-print

We convert the online

arXiv:2103.04543v2
fatcat:r54ordqp7vc37f6kmsrhfvzjqe
*spanner**problem**and*the online Steiner forest*problem*into online*covering**problems**and*round in a*problem*-specific fashion. ... We present online algorithms*for*directed*spanners**and*Steiner forests. ... Acknowledgements We thank the anonymous reviewers*for*comments*and*suggestions that helped improve the presentation. ...##
###
The Minimum Edge Compact Spanner Network Design Problem
[article]

2017
*
arXiv
*
pre-print

We prove hardness results

arXiv:1712.04010v1
fatcat:73s4wkxusnhddk6nva3bu52md4
*related*to the*problem*, design exact*and*greedy algorithms*for*solving the*problem*,*and*show*related*experimental results. ... In this paper we introduce*and*study the Minimum Edge Compact*Spanner*~(MECS)*problem*. ... A*related*class of*problems*is that of Network Design*and*have been studied*for*a long time. The sparse*spanner**problem*is a special instance of a network design*problem*. ...##
###
Small Cuts and Connectivity Certificates: A Fault Tolerant Approach

2019
*
International Symposium on Distributed Computing
*

., with O(1) rounds)

doi:10.4230/lipics.disc.2019.30
dblp:conf/wdag/Parter19
fatcat:xjsv2bvopjdofpmaolxhrczcvy
*for**problems*that are closely*related*to hard global*problems*(i.e., with a lower bound of Ω(Diam + √ n) rounds). ...*and*an integer λ, a λ-sparse certificate H is a*subgraph*H ⊆ G with O(λn) edges which is λ-connected iff G is λ-connected. ... The*relation*between FT-*spanners**and*connectivity certificates is based on the following observation. Observation 25. ...##
###
Approximation of minimum weight spanners for sparse graphs

2011
*
Theoretical Computer Science
*

The sparsest t-

doi:10.1016/j.tcs.2010.11.034
fatcat:r24q6gfu5be3djl7rwqhiyqfwy
*spanner**problem*asks to find,*for*a given graph G*and*an integer t, a t-*spanner*of G with the minimum number of edges. ...*For*t ≥ 5, the*problem*remains NP-hard*for*planar graphs*and*the approximability status of the*problem*on planar graphs was open. ... Acknowledgements We would like to thank the anonymous referees*for*comments which improved the presentation of this paper. ...
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