Set connectivity problems in undirected graphs and the directed steiner network problem

Chekuri, Chandra; Even, Guy; Gupta, Anupam; Segev, Danny

doi:10.1145/1921659.1921664

Cited by 53 publications

(84 citation statements)

References 21 publications

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“…Halperin and Krauthgamer [88] proved that an O(log 2−ε n) approximation algorithm for directed Steiner tree for some constant ε > 0 would imply that NP has randomized quasi-polynomial time algorithms; on the positive side, if there are k terminals to be connected to the root, the best known algorithm for the problem gives an approximation ratio of ε −3 k ε and runs in time n O(1/ε) [27]. The directed Steiner forest problem is hard to approximate better than Ω(2 log 1−ε n ) for any ε > 0, via a reduction from the Label Cover problem [49]; the current best algorithms gives an approximation guarantee of O(k 1/2+ε ) [32] and n ε · min{n 4/5 , m 2/3 } [57] (note that these two results are incomparable since k could be as large as Θ(n 2 )).…”

Section: Other Ideas and Techniquesmentioning

confidence: 99%

Approximation algorithms for network design: A survey

Gupta

Könemann

2011

Surveys in Operations Research and Management Science

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In a typical instance of a network design problem, we are given a directed or undirected graph G = (V, E), non-negative edge-costs c e for all e ∈ E, and our goal is to find a minimum-cost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimum-cost set of edges that induces a connected graph (this is the minimum-cost spanning tree problem), or we might want to find a minimum-cost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimum-cost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples.Many practically relevant instances of network design problems are NP-hard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomial-time), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an α-approximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the following we will also sometimes refer to α as the performance guarantee of the respective approximation algorithm.The last 30 years have seen a tremendous amount of research on approximation algorithms for network design problems. And over this period, several technical themes have emerged, and have been explored and exploited to give algorithms and analyze their performance. Our aim in this survey is to provide an overview over these techniques. Each of the following sections focuses on one technique and has two main parts: first, we present an introductory application to the well-known classical minimum-spanning tree problem. The second part of each section demonstrates more sophisticated recent example applications of the respective technique. Throughout we assume that the reader is familiar with fundamental concepts of graph theory, combinatorial optimization, and approximation algorithms. While we may recap certain key definitions, we rely on the reader to be familiar with others. We refer to the excellent text books [45,160,163] for background reading.The minimum spanning tree problem has been studied for at least a century, and it is clearly one of the most prominent network design problems. The input to an instance of this problem consists of an undirected graph G = (V, E) each of whose edges e ∈ E is endowed by an arbitrary cost c e , and the goal is to compute a spanning tree of smallest cost. The earliest known algorithm for this problem was developed by Boruvka [21], and since then a vast number of techniques have...

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Section: Other Ideas and Techniquesmentioning

confidence: 99%

Approximation algorithms for network design: A survey

Gupta

Könemann

2011

Surveys in Operations Research and Management Science

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“…We exploit the known upper bound of 2 on the integrality gap of a natural LP for the Survivable Network Design Problem with vertex connectivity requirements in {0, 1, 2} [16]. The bucketing and scaling trick has seen several uses in the past, and has recently been highlighted in [8,9].…”

Section: Overview Of Technical Ideasmentioning

confidence: 99%

“…Recall that the was defined as follows: Given a graph G(V , E) with edge-costs, a set T ⊆ V \ {r} of terminals, and a root r ∈ V (G), find a subgraph H of minimum density, in which every terminal of H is 2-connected to r. We describe an algorithm for DENS-2VC that gives an O(log )-approximation, where = |T | is the number of terminals. We use an LP based approach and a bucketing and scaling trick (see [8,9] for applications of this idea), and a constant-factor bound on the integrality gap of an LP for SNDP with vertex-connectivity requirements in {0, 1, 2} [16].…”

Section: An O(log )-Approximation For the Dens-2vc Problemmentioning

confidence: 99%

Pruning 2-Connected Graphs

Chekuri

Korula

2010

Algorithmica

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Given an undirected graph G with edge costs and a specified set of terminals, let the density of any subgraph be the ratio of its cost to the number of terminals it contains. If G is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of G? We answer this question in the affirmative by giving an algorithm to prune G and find such subgraphs of any desired size, incurring only a logarithmic factor increase in density (plus a small additive term).We apply our pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimumcost 2-vertex-connected subgraph of G containing at least k vertices. In the Budget-2VC problem, we are given a graph G with edge costs, and a budget B; the goal is to find a 2-vertex-connected subgraph H of G with total edge cost at most B that maximizes the number of vertices in H . We describe an O(log n log k) approximation for the k-2VC problem, and a bicriteria approximation for the Budget-2VC problem that gives an O( 1 log 2 n) approximation, while violating the budget by a factor of at most 2 + .

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“…This gives an O(i 3 ·k 1/i )-approximation algorithm in time O(n i ). This method is applicable to both EW-GST and NW-GST, but again, could not yield a polynomial-time polylogarithmic approximation algorithm (See, e.g., [20,9]). …”

Section: Introductionmentioning

confidence: 99%

Beyond Metric Embedding: Approximating Group Steiner Trees on Bounded Treewidth Graphs

Chalermsook¹,

Das²,

Laekhanukit³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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The Group Steiner Tree (GST) problem is a classical problem in combinatorial optimization and theoretical computer science. In the Edge-Weighted Group Steiner Tree (EW-GST) problem, we are given an undirected graph G = (V, E) on n vertices with edge costs c : E → R ≥0 , a source vertex s and a collection of subsets of vertices, called groups, S1, . . . , S k ⊆ V . The goal is to find a minimum-cost tree H ⊆ G that connects s to some vertex from each group Si, for all i = 1, 2, . . . , k. The Node-Weighted Group Steiner Tree (NW-GST) problem has the same setting, but the costs are associated with nodes. The goal is to find a minimumcost node set X ⊆ V such that G[X] connects every group to the source.When G is a tree, both EW-GST and NW-GST admit a polynomial-time O(log n log k) approximation algorithm due to the seminal result of [Garg et al., SODA'98 and J. Algorithm]. The matching hardness of log 2− n is known even for tree instances of EW-GST and NW-GST [Halperin and Krauthgamer STOC'03]. In general graphs, most of polynomial-time approximation algorithms for EW-GST reduce the problem to a tree instance using the metrictree embedding, incurring a loss of O(log n) on the approximation factor [Bartal, FOCS'96; Fakcharoenphol et al., FOCS'03 and JCSS]. This yields an approximation ratio of O(log 2 n log k) for EW-GST. Using metric-tree embedding, this factor cannot be improved: The loss of Ω(log n) is necessary on some input graphs (e.g., grids and expanders). There are alternative approaches that avoid metric-tree embedding, e.g., the algorithm of [Chekuri and Pal, FOCS'05], which gives a tight approximation ratio, but none of which achieves polylogarithmic approximation in polynomial-time. This state of the art shows a clear lack of understanding of GST in general graphs beyond the metric-tree embedding technique. For NW-GST (for which the metric-tree embedding does not apply), not even a polynomial-time polylogarithmic approximation algorithm is known.In this paper, we present O(log n log k) approximation algorithms that run in time nÕ2 ) for both NW-GST * Aalto University, Finland. Work partially done while at Max-Planck-Institut für Informatik, Germany, email: parinya.chalermsook@aalto.fi † Universität Bremen, Germany.

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Set connectivity problems in undirected graphs and the directed steiner network problem

Cited by 53 publications

References 21 publications

Approximation algorithms for network design: A survey

Approximation algorithms for network design: A survey

Pruning 2-Connected Graphs

Beyond Metric Embedding: Approximating Group Steiner Trees on Bounded Treewidth Graphs

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