2-Node-Connectivity Network Design

Nutov, Zeev

doi:10.1007/978-3-030-80879-2_15

Cited by 10 publications

(21 citation statements)

References 39 publications

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“…Besides giving a 1.892 approximation for 1-Node-CAP, which improves upon the state-of-the-art approximation of 1.91 by Nutov [27] for Block-TAP, one key point of our work is that the analysis of our approximation bound is quite simple compared to the existing results in the literature for CacAP that achieve a better than 2 approximation. More precisely, only the algorithm of [6] gives a better approximation factor than our algorithm-in fact, much better -but its analysis is far more involved, spanning over 70 pages.…”

Section: Introductionmentioning

confidence: 62%

“…In general, the Node Steiner Tree problem is as hard to approximate as the Set Cover problem, and it admits a O(log |R|) approximation algorithm (which holds even in the more general weighted version [22]). However, the instances that arise from the reductions of [2,4,27] have special properties that allow for a constant factor approximation. We now define the properties of these instances.…”

Section: Reduction To Ca-node-steiner-tree Instancesmentioning

confidence: 99%

“…Similar to CacAP, a 2-approximation algorithm was known for quite some time [17,21] for Block-TAP, and until recently, it was an open question to design an approximation algorithm with ratio better than 2. Very recently, Nutov [27] observed that Block-TAP can be reduced to special instances of the (unweighted) Node Steiner Tree problem, extending the techniques of Basavaraju et al [2] and Byrka et al [4] used for CacAP. We recall that the (unweighted) Node Steiner Tree problem takes as input a graph G and a subset of nodes (called terminals), and asks to find a tree spanning the terminals which minimizes the number of nonterminal nodes included in the tree.…”

Section: Introductionmentioning

confidence: 95%

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Node Connectivity Augmentation via Iterative Randomized Rounding

Angelidakis¹,

Hyatt-Denesik²,

Sanità³

2021

Preprint

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Many fundamental network design problems deal with the design of low-cost networks that are resilient to the failure of their elements (such as nodes or links). One such problem is Connectivity Augmentation, where the goal is to cheaply increase the connectivity of a given network from a value k to k + 1.The most studied setting focuses on edge-connectivity, and it is well-known that it can be reduced to the case k = 2, called Cactus Augmentation. From an approximation perspective, Byrka, Grandoni, and Jabal Ameli (2020) were the first who managed to break the 2-approximation barrier for this problem, by exploiting a connection to the Steiner Tree problem, and by tailoring the analysis of the iterative randomized rounding technique for Steiner Tree to the specific instances arising from this connection. Very recently, Nutov (2020) observed that a similar reduction to Steiner Tree also holds for a node-connectivity problem called Block-Tree Augmentation, where the goal is to add edges to a given spanning tree in order to make the resulting graph 2-node-connected. Block-Tree Augmentation can be seen as the direct generalization of the well-studied Tree Augmentation problem (k = 1) to the node-setting. Combining Nutov's result with the algorithm of Byrka, Grandoni, and Jabal Ameli yields a 1.91-approximation for Block-Tree Augmentation, that is the best bound known so far.In this work, we give a 1.892-approximation algorithm for the problem of augmenting the nodeconnectivity of any given graph from 1 to 2. As a corollary, we improve upon the state-of-the-art approximation factor for Block-Tree Augmentation. Our result is obtained by developing a different and simpler analysis of the iterative randomized rounding technique when applied to the Steiner Tree instances arising from the aforementioned reductions. Our results also imply a 1.892-approximation algorithm for Cactus Augmentation. While this does not beat the best approximation factor by Cecchetto, Traub, and Zenklusen (2021) that is known for this problem, a key point of our work is that the analysis of our approximation factor is quite simple compared to previous results in the literature. In addition, our work gives new insights on the iterative randomized rounding method, that might be of independent interest.

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Section: Introductionmentioning

confidence: 62%

Section: Reduction To Ca-node-steiner-tree Instancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

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Node Connectivity Augmentation via Iterative Randomized Rounding

Angelidakis¹,

Hyatt-Denesik²,

Sanità³

2021

Preprint

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“…For the case of vertexconnectivity augmentation of a 2-connected graph, Auletta et al [2] provide a 2-approximation, while for the case of k-connected graphs the current best approximation ratio is 4 + ε for any fixed k, due to Nutov [26]. Recently, for the case of vertex-connectivity augmentation of 1-connected graphs, Nutov developed a 1.91-approximation [25].…”

Section: Related Resultsmentioning

confidence: 99%

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

Gálvez¹,

Sanhueza-Matamala²,

Soto³

2021

Preprint

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Given a k-vertex-connected graph G and a set S of extra edges (links), the goal of the kvertex-connectivity augmentation problem is to find a set S ⊆ S of minimum size such that adding S to G makes it (k + 1)-vertex-connected. Unlike the edge-connectivity augmentation problem, research for the vertex-connectivity version has been sparse.In this work we present the first polynomial time approximation algorithm that improves the known ratio of 2 for 2-vertex-connectivity augmentation, for the case in which G is a cycle. This is the first step for attacking the more general problem of augmenting a 2-connected graph.Our algorithm is based on local search and attains an approximation ratio of 1.8704. To derive it, we prove novel results on the structure of minimal solutions. * Full version of the extended abstract accepted at WAOA 2021. In that extended abstract an approximation ratio of 1.8703 was claimed, but it has been corrected to 1.8704 in this version. We apologize for the inconvenience.

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“…Nutov [15] recently presented a 1.91-approximation algorithm for unweighted 2NC-TAP, thus beating the threshold of 2; this result does not use any LP relaxation.…”

Section: Introductionmentioning

confidence: 99%

On a Partition LP Relaxation for Min-Cost 2-Node Connected Spanning Subgraphs

Grout¹,

Cheriyan²,

Laekhanukit³

2021

Preprint

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Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints.The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation.

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2-Node-Connectivity Network Design

Cited by 10 publications

References 39 publications

Node Connectivity Augmentation via Iterative Randomized Rounding

Node Connectivity Augmentation via Iterative Randomized Rounding

Approximation Algorithms for Vertex-Connectivity Augmentation on the Cycle

On a Partition LP Relaxation for Min-Cost 2-Node Connected Spanning Subgraphs

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