Approximation Algorithms for the Steiner Tree Problem in Graphs

Gröpl, Clemens; Hougardy, Stefan; Nierhoff, Till; PrOmel, Hans Jtirgen

doi:10.1007/978-1-4613-0255-1_7

Cited by 56 publications

(41 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now we need the following lemma, whose proof is obtained from Robins-Zelikovsky as presented in [11] by changing what quantities represent and some parameters, together with a "fractional cover" idea from the submitted journal version of [3]. Lemma 5 Assuming that for the minimum spanning tree T , constant 0 < α < 1, and for any collection of stars B, there exist non-negative coefficients (x S ) such that S x S f B (S) ≥ c(T ) − f (B) and S x S p(S) ≤ αopt, where opt is the power of the optimum solution, the Greedy algorithms' output has power at most βopt where β = 1 + α + α ln(1/α).…”

Section: Lemma 4 Let B Be An Arbitrary Collection Of Stars and T Be mentioning

confidence: 99%

Approximate Min-Power Strong Connectivity

Călinescu¹

2013

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Given a directed simple graph G = (V, E) and a cost function c : E → R + , the power of a vertex u in a directed spanning subgraph H is given by p H (u) = max uv∈E(H) c(uv), and corresponds to the energy consumption required for wireless node u to transmit to all nodes v with uv ∈ E(H). The power of H is given by p(H) = u∈V p H (u).Power Assignment seeks to minimize p(H) while H satisfies some connectivity constraint. In this paper, we assume E is bidirected (for every directed edge e ∈ E, the opposite edge exists and has the same cost), while H is required to be strongly connected. This is the original power assignment problem introduced by Chen and Huang in 1989, who proved that a bidirected minimum spanning tree has approximation ratio at most 2 (this is tight). In Approx 2010, we introduced a greedy approximation algorithm and claimed a ratio of 1.992. Here we improve the algorithm's analysis to 1.85, combining techniques from Robins-Zelikovsky (2000) for Steiner Tree, and Caragiannis, Flammini, and Moscardelli (2007) for the broadcast version of Power Assignment, together with a simple idea inspired by Byrka, Grandoni, Rothvoß, and Sanità (2010).The proof also shows that a natural linear programming relaxation, introduced by Calinescu and Qiao in Infocom 2012, has integrality gap at most 1.85.

show abstract

Section: Lemma 4 Let B Be An Arbitrary Collection Of Stars and T Be mentioning

confidence: 99%

Approximate Min-Power Strong Connectivity

Călinescu¹

2013

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…It is NP-hard, and has been studied intensively from an approximation viewpoint (see [18] for a survey on these results). The best known ratio obtained so far is 1 + ln(3)/2 ≃ 1.55 [30].…”

Section: Results On Some Particular Problemsmentioning

confidence: 99%

Complexity and Approximation in Reoptimization

Ausiello

Bonifaci

Escoffier

2011

Computability in Context

View full text Add to dashboard Cite

Abstract. In this survey the following model is considered. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance I. How can we exploit the knowledge we have on the solution of instance I to compute a (approximate) solution of instance I ′ in an efficient way? This computation model is called reoptimization and is of practical interest in various circumstances. In this article we first discuss what kind of performance we can expect for specific classes of problems and then we present some classical optimization problems (i.e. Max Knapsack, Min Steiner Tree, Scheduling) in which this approach has been fruitfully applied. Subsequently, we address vehicle routing problems and we show how the reoptimization approach can be used to obtain good approximate solution in an efficient way for some of these problems.

show abstract

“…Using the same proof arguments given in [3], we can prove the following approximation bound of 3-Restricted Greedy Algorithm.…”

Section: -Restricted Greedy Algorithmmentioning

confidence: 92%

Approximations for node-weighted Steiner tree in unit disk graphs

Wang

et al. 2010

Optim Lett

View full text Add to dashboard Cite

Given a node-weighted connected graph and a subset of terminals, the problem node-weighted Steiner tree (NWST) seeks a lightest tree connecting a given set of terminals in a node-weighted graph. While NWST in general graphs are as hard as Set Cover, NWST restricted to unit-disk graphs (UDGs) admits COCOA, 2008, pp 278-285) showed that any μ-approximation algorithm for the classical edge-weighted Steiner tree problem can be used to produce 2.5 μ-approximation algorithm for NWST restricted to UDGs. With the best known approximation bound 1.55 for the classical Steiner tree problem, they obtained an approximation bound 3.875 for NWST restricted to UDGs. In this paper, we present three approximation algorithms for NWST restricted to UDGs, the k-Restricted Relative Greedy Algorithm whose approximation bound converges to 1 + ln 5 ≈ 2. 61 as k → ∞, the 3-Restricted Greedy Algorithm with approximation bound 4 1 3 , and the k-Restricted Variable Metric Algorithm whose approximation bound converges to 3.9334 as k → ∞.

show abstract

Approximation Algorithms for the Steiner Tree Problem in Graphs

Cited by 56 publications

References 19 publications

Approximate Min-Power Strong Connectivity

Approximate Min-Power Strong Connectivity

Complexity and Approximation in Reoptimization

Approximations for node-weighted Steiner tree in unit disk graphs

Contact Info

Product

Resources

About