# Approximation algorithms for metric tree cover and generalized tour and tree covers

• Volume: 41, Issue: 3, page 305-315
• ISSN: 0399-0559

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## Abstract

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Given a weighted undirected graph G = (V,E), a tree (respectively tour) cover of an edge-weighted graph is a set of edges which forms a tree (resp. closed walk) and covers every other edge in the graph. The tree (resp. tour) cover problem is of finding a minimum weight tree (resp. tour) cover of G. Arkin, Halldórsson and Hassin (1993) give approximation algorithms with factors respectively 3.5 and 5.5. Later Könemann, Konjevod, Parekh, and Sinha (2003) study the linear programming relaxations and improve both factors to 3. We describe in the first part of the paper a 2-approximation algorithm for the metric case of tree cover. In the second part, we will consider a generalized version of tree (resp. tour) covers problem which is to find a minimum tree (resp. tours) which covers a subset D ⊆ E of G. We show that the algorithms of Könemann et al. can be adapted for the generalized tree and tours covers problem with the same factors.

## How to cite

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Nguyen, Viet Hung. "Approximation algorithms for metric tree cover and generalized tour and tree covers." RAIRO - Operations Research 41.3 (2007): 305-315. <http://eudml.org/doc/250087>.

@article{Nguyen2007,
abstract = { Given a weighted undirected graph G = (V,E), a tree (respectively tour) cover of an edge-weighted graph is a set of edges which forms a tree (resp. closed walk) and covers every other edge in the graph. The tree (resp. tour) cover problem is of finding a minimum weight tree (resp. tour) cover of G. Arkin, Halldórsson and Hassin (1993) give approximation algorithms with factors respectively 3.5 and 5.5. Later Könemann, Konjevod, Parekh, and Sinha (2003) study the linear programming relaxations and improve both factors to 3. We describe in the first part of the paper a 2-approximation algorithm for the metric case of tree cover. In the second part, we will consider a generalized version of tree (resp. tour) covers problem which is to find a minimum tree (resp. tours) which covers a subset D ⊆ E of G. We show that the algorithms of Könemann et al. can be adapted for the generalized tree and tours covers problem with the same factors. },
author = {Nguyen, Viet Hung},
journal = {RAIRO - Operations Research},
keywords = {Approximation algorithms; graph algorithms; network design; approximation algorithms},
language = {eng},
month = {8},
number = {3},
pages = {305-315},
publisher = {EDP Sciences},
title = {Approximation algorithms for metric tree cover and generalized tour and tree covers},
url = {http://eudml.org/doc/250087},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Nguyen, Viet Hung
TI - Approximation algorithms for metric tree cover and generalized tour and tree covers
JO - RAIRO - Operations Research
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 305
EP - 315
AB - Given a weighted undirected graph G = (V,E), a tree (respectively tour) cover of an edge-weighted graph is a set of edges which forms a tree (resp. closed walk) and covers every other edge in the graph. The tree (resp. tour) cover problem is of finding a minimum weight tree (resp. tour) cover of G. Arkin, Halldórsson and Hassin (1993) give approximation algorithms with factors respectively 3.5 and 5.5. Later Könemann, Konjevod, Parekh, and Sinha (2003) study the linear programming relaxations and improve both factors to 3. We describe in the first part of the paper a 2-approximation algorithm for the metric case of tree cover. In the second part, we will consider a generalized version of tree (resp. tour) covers problem which is to find a minimum tree (resp. tours) which covers a subset D ⊆ E of G. We show that the algorithms of Könemann et al. can be adapted for the generalized tree and tours covers problem with the same factors.
LA - eng
KW - Approximation algorithms; graph algorithms; network design; approximation algorithms
UR - http://eudml.org/doc/250087
ER -

## References

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1. E.M. Arkin, M.M. Halldórsson and R. Hassin, Approximating the tree and tour covers of a graph. Inf. Process. Lett.47 (1993) 275–282.  Zbl0785.68041
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5. Y.J. Chu and T.H. Liu, On the shortest arborescence of a directed graph. Scientia Sinica14 (1965).  Zbl0178.27401
6. J. Könemann, G. Konjevod, O. Parekh and A. Sinha, Improved approximations for tour and tree covers. Algorithmica38 (2003) 441–449.  Zbl1138.68666
7. D.B. Shmoys and D.P. Williamsons, Analyzing the help-karp tsp bound: a monotonicity property with application. Inf. Process. Lett.35 (1990) 281–285.
8. V.V. Vazirani and S. Rajagopalan, On the bidirected cut relaxation for metric bidirected steiner tree problem, in Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (1999) 742–751.  Zbl0938.68072
9. L.A. Wolsey, Heuristic analysis, linear programming and branch-and-bound. Math. Program. Stud.13 (1980) 121–134.  Zbl0442.90061

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