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

RAIRO - Operations Research (2007)

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

## Access Full Article

top## Abstract

top## How to cite

topNguyen, 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

top- 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.
- J. Edmonds, Optimum branchings. J. Res. Nat. Bur. Stand. B71 (1965).
- M.R. Garey and D.S. Johnson, Computer and Intractablity: A Guide to the theory of the NP-Completeness, Freeman (1978).
- M.X. Goemans and D.J. Bertsimas, Survivable networks, linear programming relaxations and the parsinomious property. Math. Program.60 (1993) 145–166.
- Y.J. Chu and T.H. Liu, On the shortest arborescence of a directed graph. Scientia Sinica14 (1965).
- J. Könemann, G. Konjevod, O. Parekh and A. Sinha, Improved approximations for tour and tree covers. Algorithmica38 (2003) 441–449.
- 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.
- 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.
- L.A. Wolsey, Heuristic analysis, linear programming and branch-and-bound. Math. Program. Stud.13 (1980) 121–134.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.