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

Dagmar Handke, Guy Kortsarz
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

Jan Vondrák
2006 Random structures & algorithms (Print)
With the goal of speeding up the repeated queries 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]

András Sebő, Eric Tannier
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

Dagmar Handke
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]

Michael Elkin, David Peleg
2001 Lecture Notes in Computer Science
of the 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

Michael Elkin, David Peleg
2005 Theoretical Computer Science
of the 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

Feodor F. Dragan, Fedor V. Fomin, Petr A. Golovach
2011 Journal of computer and system sciences (Print)
In particular, 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]

Feodor F. Dragan, Fedor V. Fomin, Petr A. Golovach
2008 Lecture Notes in Computer Science
In particular, 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

Leizhen Cai, Derek G. Corneil
1995 SIAM Journal on Discrete Mathematics
Furthermore, linear-time algorithms are derived 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]

Elena Grigorescu, Young-San Lin, Kent Quanrud
2021 arXiv   pre-print
We convert the online 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]

Tathagata Mukherjee, Alexander Veremyev, Piyush Kumar, Eduardo Pasiliao Jr
2017 arXiv   pre-print
We prove hardness results 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

Merav Parter, Michael Wagner
2019 International Symposium on Distributed Computing
., with O(1) rounds) 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

Feodor F. Dragan, Fedor V. Fomin, Petr A. Golovach
2011 Theoretical Computer Science
The sparsest t-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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