When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks

Agrawal, Ajit; Klein, Philip N.; Ravi, R.

doi:10.1137/s0097539792236237

Cited by 339 publications

(581 citation statements)

References 30 publications

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“…The obvious question left open by [12], that we answer affirmatively in this paper The algorithms in [1,9] are based on the classical undirected cut formulation for Steiner forests [2]. The integrality gap of this relaxation is known to be (2 − 1/k) and the results in [1,9] are therefore tight.…”

Section: Introductionmentioning

confidence: 66%

“…The integrality gap of this relaxation is known to be (2 − 1/k) and the results in [1,9] are therefore tight. Our lifted-cut dual relaxation is strictly stronger than the classical undirected cut formulation.…”

Section: Introductionmentioning

confidence: 83%

“…The algorithm is similar to the primal-dual algorithm AKR due to Agrawal, Klein, and Ravi [1]. In this section, we first state the standard LP formulation on which AKR is based, then review AKR, and finally describe the cost sharing algorithm KLS from [12].…”

Section: Cross-monotonic Cost Sharing Methods For Steiner Forestsmentioning

confidence: 99%

“…The best known algorithm for the Steiner forest problem due to Agrawal, Klein and Ravi [1], and Goemans and Williamson [9] use the primal-dual schema. The algorithms in [1,9] achieve an approximation ratio of (2 − 1/k).…”

Section: Introductionmentioning

confidence: 99%

“…The authors present a cost sharing method KLS which is an adaptation of the primal-dual algorithm AKR for Steiner forests due to Agrawal, Klein, and Ravi [1]. The cost shares computed by KLS are proven to be 2-budget balanced.…”

Section: Introductionmentioning

confidence: 99%

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From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem

Könemann

Leonardi

Schäfer

et al. 2005

Automata, Languages and Programming

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In this paper we consider a game theoretical variant of the Steiner forest problem. An instance of this game consists of an undirected graph G = (V, E), non-negative costs c(e) for all edges e in E, and k players. Each player i has an associated pair of terminals s i and t i . Consider a forest F in G. We say that player i is serviced if s i and t i are connected in F. Player i derives a private utility u i for receiving service. In a recent paper, Könemann, Leonardi, and Schäfer [12] showed that a natural primal-dual algorithm, KLS, gives rise to a 2-approximate budget balanced and group-strategyproof cost sharing method for the above game.In this paper we show that the techniques used in [12] yield a new linear programming relaxation for the Steiner forest problem: the lifted-cut relaxation. First, we give an alternate proof of the approximate budget-balance result in [12] by showing that the cost shares computed by algorithm KLS are feasible for the dual of this relaxation. Second, we are able to show that this new undirected relaxation for Steiner forests is strictly stronger than the well-studied undirected cut relaxation.We conclude the paper with a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree and Steiner forest games. This shows that the results of [11,12] are essentially tight.

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Section: Introductionmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 83%

Section: Cross-monotonic Cost Sharing Methods For Steiner Forestsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem

Könemann

Leonardi

Schäfer

et al. 2005

Automata, Languages and Programming

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Algorithms for a network design problem with crossing supermodular demands

Melkonian

Tardos

2004

Networks

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We present approximation algorithms for a class of directed network design problems. The network design problem is to find a minimum cost subgraph such that for each vertex set S there are at least f(S) arcs leaving the set S. In the last 10 years general techniques have been developed for designing approximation algorithms for undirected network design problems. Recently, Kamal Jain gave a 2-approximation algorithm for the case when the function f is weakly supermodular. There has been very little progress made on directed network design problems. The main techniques used for the undirected problems do not have simple extensions to the directed case. Andrá s Frank has shown that in a special case when the function f is intersecting supermodular the problem can be solved optimally. In this article, we use this result to get a 2-approximation algorithm for a more general case when f is crossing supermodular. We also extend Jain's techniques to directed problems. We prove that if the function f is crossing supermodular, then any basic solution of the LP relaxation of our problem contains at least one variable with value greater or equal to 1 4 . This result implies a 4-approximation algorithm for the class of directed network design problems.

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Approximation algorithms for finding low‐degree subgraphs

Klein¹,

Krishnan²,

Raghavachari³

et al. 2004

Networks

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We give quasipolynomial-time approximation algorithms for designing networks with a minimum degree. Using our methods, one can design networks whose connectivity is specified by "proper" functions, a class of 0 -1 functions indicating the number of edges crossing each cut. We also provide quasipolynomial-time approximation algorithms for finding two-edge-connected spanning subgraphs of approximately minimum degree of a given two-edge-connected graph, and a spanning tree (branching) of approximately minimum degree of a directed graph. The degree of the output network in all cases is guaranteed to be at most (1 ؉ ⑀) times the optimal degree, plus an additive O(log 1؉⑀ n) for any ⑀ > 0. Our analysis indicates that the degree of an optimal subgraph for each of the problems above is well estimated by certain polynomially solvable linear programs. This suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch and bound.

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When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks

Cited by 339 publications

References 30 publications

From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem

From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem

Algorithms for a network design problem with crossing supermodular demands

Approximation algorithms for finding low‐degree subgraphs

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