Primal-Dual Approaches to the Steiner Problem

Polzin, Tobias; Daneshmand, Siavash Vahdati

doi:10.1007/3-540-44436-x_22

Cited by 8 publications

(5 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Raghavan [194] also studied and computationally compared four efficient implementations of Wong's dual‐ascent algorithm for solving the SAP. Even though the dual‐ascent algorithm applied to the strong MCF/DCUT formulation generally performs well in practice, its lower bounds depend on the choice of the root node (see Polzin and Vahdati Daneshmand [181] for an example). Moreover, the solution of Wong's dual‐ascent algorithm can be arbitrarily far from the optimum (Vahdati Daneshmand [228] shows this result for undirected graphs and Candia‐Véjar and Bravo‐Azlán [38] prove this result for directed graphs).…”

Section: Dual‐ascent Methodsmentioning

confidence: 99%

“…On directed graphs, however, Raghavan's implementation effectively changes the rate at which the dual variables are increased on different arcs, as an arc can be in multiple cut‐sets. Several years later, Polzin and Vahdati Daneshmand [181] independently provided an implementation very similar to the one given by Raghavan [194], proving that it guarantees the worst‐case approximation ratio of 2 − 2/| R |, while providing empirically high quality bounds.…”

Section: Dual‐ascent Methodsmentioning

confidence: 99%

“…Construction heuristics which are much easier to implement and are therefore frequently used in computational frameworks (cf. Sections 7 and 8) like the TM heuristic [218], the distance network heuristic [170], the primal‐dual algorithm [111], the dual‐ascent approach from Polzin and Vahdati Daneshmand [181], or the local search from Uchoa and Werneck [226] yield (2 − 2/| R |)‐approximations. Despite the fact that the DCUT model is known to be stronger than its undirected counterpart [109] no better LP‐based approximation algorithm based on this formulation is known.…”

Section: Approximation Algorithmsmentioning

confidence: 99%

See 2 more Smart Citations

Solving Steiner trees: Recent advances, challenges, and perspectives

Ljubić

2020

Networks

View full text Add to dashboard Cite

The Steiner tree problem (STP) in graphs is one of the most studied problems in combinatorial optimization. Since its inception in 1970, numerous articles published in the journal Networks have stimulated new theoretical and computational studies on Steiner trees: from approximation algorithms, heuristics, metaheuristics, all the way to exact algorithms based on (mixed) integer linear programming, fixed parameter tractability, or combinatorial branch‐and‐bounds. The pervasive applicability and relevance of Steiner trees have been reinforced by the recent 11th DIMACS Implementation Challenge in 2014 and the PACE 2018 Challenge. This article provides an overview of the rich developments from the last three decades for the STP in graphs and highlights the most recent computational studies for some of its closely related variants.

show abstract

Section: Dual‐ascent Methodsmentioning

confidence: 99%

Section: Dual‐ascent Methodsmentioning

confidence: 99%

Section: Approximation Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

Solving Steiner trees: Recent advances, challenges, and perspectives

Ljubić

2020

Networks

View full text Add to dashboard Cite

show abstract

“…Thus our framework combines these two decomposition approaches. Furthermore, due to similarities between the SSTP and its deterministic variant, a third option for computing lower bounds appears promising: a dual ascent procedure, which constructs an initial dual solution in a greedy scheme (see [28, 30, 40]). For the STP and its variants, such procedures are known empirically to obtain high quality bounds, which in some cases are even tight enough to solve the corresponding problem to optimality in a branch-and-bound (B&B) procedure [26, 28, 31].…”

Section: Algorithmic Frameworkmentioning

confidence: 99%

Decomposition methods for the two-stage stochastic Steiner tree problem

et al. 2017

View full text Add to dashboard Cite

A new algorithmic approach for solving the stochastic Steiner tree problem based on three procedures for computing lower bounds (dual ascent, Lagrangian relaxation, Benders decomposition) is introduced. Our method is derived from a new integer linear programming formulation, which is shown to be strongest among all known formulations. The resulting method, which relies on an interplay of the dual information retrieved from the respective dual procedures, computes upper and lower bounds and combines them with several rules for fixing variables in order to decrease the size of problem instances. The effectiveness of our method is compared in an extensive computational study with the state-of-the-art exact approach, which employs a Benders decomposition based on two-stage branch-and-cut, and a genetic algorithm introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms existing ones, both on benchmark instances from literature, as well as on large-scale telecommunication networks.

show abstract

“…This relaxation has an integrality gap of 2 − 2 |R| and the analysis of these algorithms is therefore tight. Slightly improved algorithms have since been designed [23,26] but do not achieve any constant approximation factor better than 2.…”

Section: Approaches Based On Linear Programsmentioning

confidence: 99%

A partition-based relaxation for Steiner trees

2009

View full text Add to dashboard Cite

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals R ⊆ V , and non-negative costs c e for all edges e ∈ E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The nodes V \R are called Steiner nodes.The best approximation algorithm known for the Steiner tree problem is due to Robins and Zelikovsky (SIAM J. Discrete Math, 2005); their greedy algorithm achieves a performance guarantee of 1 + ln 3 2 ≈ 1.55. The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math. Programming, 1993) and achieves an approximation ratio of 2 − 2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky's algorithm has a natural primal-dual interpretation with respect to a novel partition-based linear programming relaxation. We also exhibit surprising connections between the new formulation and existing LPs and we show that the new LP is stronger than the bidirected cut formulation.An instance is b-quasi-bipartite if each connected component of G\R has at most b vertices. We show that Robins' and Zelikovsky's algorithm has an approximation ratio better than 1 + ln 3 2 for such instances, and we prove that the integrality gap of our LP is between 8 7 and 2b+1 b+1 . Greedy algorithms and r-Steiner treesOne of the first approximation algorithms for the Steiner tree problem is the well-known minimum-spanning tree heuristic which is widely attributed to Moore [14]. Moore's algorithm has a performance ratio of 2 for the Steiner tree problem and this remained the best known until the 1990s, when Zelikovsky [41] suggested computing Steiner trees with a special structure, so called r-Steiner trees. Nearly all of the Steiner tree algorithms developed since then use r-Steiner trees. We now provide a formal definition.

show abstract

Primal-Dual Approaches to the Steiner Problem

Cited by 8 publications

References 19 publications

Solving Steiner trees: Recent advances, challenges, and perspectives

Solving Steiner trees: Recent advances, challenges, and perspectives

Decomposition methods for the two-stage stochastic Steiner tree problem

A partition-based relaxation for Steiner trees

Contact Info

Product

Resources

About