NOTE Improved Approximation Algorithms for Weighted 2- and 3-Vertex Connectivity Augmentation Problems

Penn, Michal; Shasha-Krupnik, Haya

doi:10.1006/jagm.1995.0800

Cited by 16 publications

(19 citation statements)

References 13 publications

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“…This improves the best previously known performance guarantee 3 [17,16]. This is done by combining certain properties of minimally k-connected graphs, certain techniques from recent approximation algorithms [13,17,16], and some new ideas and techniques.…”

mentioning

confidence: 90%

“…For a general instance of the minimum weight k-connected subgraph problem, approximation ratios better than in [18] were obtained for small values of k. Khuller and Raghavachari [13] developed a (2 + 1 n )-approximation algorithm for k = 2; it was improved to approximation ratio 2 in [17]. Penn and Shasha-Krupnik [17] introduced a 3-approximation algorithm for the case k = 3.…”

mentioning

confidence: 99%

“…Penn and Shasha-Krupnik [17] introduced a 3-approximation algorithm for the case k = 3. A simpler and faster 3-approximation algorithm for k = 3 was developed in [16].…”

mentioning

confidence: 99%

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A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

Auletta¹,

Dinitz²,

Nutov³

et al. 1999

Journal of Algorithms

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The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusionminimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( k 2 + 1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V | 3 |E|) = O(|V | 5 ). * Up to 1990,

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A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

Auletta¹,

Dinitz²,

Nutov³

et al. 1999

Journal of Algorithms

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“…For both the 2-ECSS and 2-VCSS problems in weighted planar graphs, the best known polynomial-time or even quasi-polynomial-time approximation guarantee is still 2 [17,21], which is achieved by polynomial-time algorithms working for general weighted graphs. On the other hand, besides the PTAS for unweighted planar graphs [6], there are better constant approximation guarantees known for general unweighted graphs.…”

Section: Introductionmentioning

confidence: 99%

Approximation Schemes for Minimum 2-Connected Spanning Subgraphs in Weighted Planar Graphs

Berger

Czumaj

Grigni

et al. 2005

Algorithms – ESA 2005

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Abstract. We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1 + ε)-spanner of G with total weight bounded by weight(A)/ε. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have non-uniform (1 or 2) connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.

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“…Most of these articles discuss specific connectivity targets (γ and/or κ have specific values) or specific topologies, like trees. Also heuristic and approximation algorithms have been proposed [85,[102][103][104][105][106][107].…”

Section: Theorem 18 the Weighted Nca Problem Is Np-hardmentioning

confidence: 99%

An Overview of Algorithms for Network Survivability

Kuipers

2012

ISRN Communications and Networking

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Network survivability-the ability to maintain operation when one or a few network components fail-is indispensable for present-day networks. In this paper, we characterize three main components in establishing network survivability for an existing network, namely, (1) determining network connectivity, (2) augmenting the network, and (3) finding disjoint paths. We present a concise overview of network survivability algorithms, where we focus on presenting a few polynomial-time algorithms that could be implemented by practitioners and give references to more involved algorithms.

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NOTE Improved Approximation Algorithms for Weighted 2- and 3-Vertex Connectivity Augmentation Problems

Cited by 16 publications

References 13 publications

A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph

Approximation Schemes for Minimum 2-Connected Spanning Subgraphs in Weighted Planar Graphs

An Overview of Algorithms for Network Survivability

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