Approximating Minimum Cost Connectivity Problems via Uncrossable Bifamilies and Spider-Cover Decompositions

Nutov, Zeev

doi:10.1109/focs.2009.9

Cited by 31 publications

(29 citation statements)

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“…Our algorithm is combinatorial, and our proof of Theorem 1.1 is relatively simple while being self contained. Moreover, the algorithm presented in this paper was recently generalized by the author in [16] to achieve the currently best knowm ratio O(k 2 ) for Rooted SND. We also note that in [8], Chuzhoy and Khanna gave an O(k 3 log n)-approximation algorithm for SND with arbitrary requirements, which is the currently best known ratio for the problem.…”

Section: Survivable Network Design (Snd)mentioning

confidence: 99%

A note on Rooted Survivable Networks

Nutov

2009

Information Processing Letters

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The (undirected) Rooted Survivable Network Design (Rooted SND) problem is: given a complete graph on node set V with edge-costs, a root s ∈ V , and (node-)connectivity requirements {r(t) : t ∈ T ⊆ V }, find a minimum cost subgraph G that contains r(t) internally-disjoint st-paths for all t ∈ T . For large values of k = max t∈T r(t) Rooted SND is at least as hard to approximate as Directed Steiner Tree [Lando & Nutov, APPROX 2008]. For Rooted SND [Chuzhoy & Khanna, FOCS 08] gave recently an approximation algorithm with ratio O(k 2 log n). Independently, and using different techniques, we obtained at the same time a simpler primal-dual algorithm with the same ratio.

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Section: Survivable Network Design (Snd)mentioning

confidence: 99%

A note on Rooted Survivable Networks

Nutov

2009

Information Processing Letters

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“…In this problem, we are given an undirected graph G = (V, E), a root vertex r and a set of terminals T ; the goal is to find a minimum-cost subgraph that has k openly (vertex) disjoint paths from the root vertex r to each terminal t ∈ T . For arbitrary k, the best known approximation ratio of this problem is O(k log k) by Nutov [24], and it was shown by Cheriyan, Laekhanukit, Naves and Vetta [7] that the dependence on k cannot be taken out because the problem does not admit o(k σ )-approximation, for some (very) small constant σ > 0, unless P = NP. However, when k is larger than the number of demands (or terminals) D, a trivial D-approximation algorithm does exist and yields a better approximation ratio than the O(k log k)-approximation algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…These problems have hardness of the form k σ , where σ is a small constant that has not been calculated. (See [24,7,3,11]). The common source of hardness of these problems is the label cover problem (a.k.a., 2P1R) with parallel repetition.…”

Section: Introductionmentioning

confidence: 99%

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Laekhanukit

2013

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

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Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important task in PCPs literature as it would imply a smaller PCP with the same or stronger soundness. While this is a basic question in PCPs community, the connection between the parameters of PCPs and hardness of approximations is sometimes obscure to approximation algorithm community. In this paper, we investigate the connection between the parameters of 2P1R and the hardness of approximating the class of so-called connectivity problems, which includes as subclasses the survivable network design and (multi)cut problems. Based on recent development on 2P1R by Chan (STOC 2013) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form k σ , for some (very) small constant σ > 0, to hardness results of the form k c for some explicit constant c, where k is a connectivity parameter. In addition, we show how to convert these hardness into hardness results of the form D c , where D is the number of demand pairs (or the number of terminals). Our results are as follows.

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“…In fact, it can be shown that for edge-connectivity, this type of spider-cover cannot occur for k ≥ 2. However, this type of spider-cover occurs for k = 2 in the next section, where node-connectivity is considered, and in a related paper [27] by the author for so-called element-connectivity. The example in Figure 4(b) shows that (for edge-connectivity) spider-covers of the second type as in Figure 4(b) can occur for k = 2, even for laminar set-families.…”

Section: Algorithm Formentioning

confidence: 99%

“…• Does NWSFC with ring-family F admit a polynomial time algorithm (under appropriate assumptions)? Note that we recently extended the results of this paper from edge-connectivity to so-called element-connectivity; see [27].…”

Section: Open Problems We Suggest the Following Open Problemsmentioning

confidence: 99%

Approximating Steiner Networks with Node-Weights

Nutov¹

2010

SIAM J. Comput.

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Abstract. The (undirected) Steiner Network problem is as follows: given a graph G = (V, E) with edge/node-weights and edge-connectivity requirements {r (u, v) : u, v ∈ U ⊆ V }, find a minimumweight subgraph H of G containing U so that the uv-edge-connectivity in H is at least r (u, v) for all u, v ∈ U . The seminal paper of Jain [Combinatorica, 21 (2001), pp. 39-60], and numerous papers preceding it, considered the Edge-Weighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum-weight edge-covers of several types of set functions and families. However, for the Node-Weighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax ·O(ln |U |), where rmax = max u,v∈U r (u, v). This generalizes the result of Klein and Ravi [J. Algorithms, 19 (1995), pp. 104-115] for the case rmax = 1. We also give an O(ln |U |)-approximation algorithm for the node-connectivity variant of NWSN (when the paths are required to be internally disjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum nodeweighted edge-cover of an uncrossable set-family. Finally, we give evidence that a polylogarithmic approximation ratio for NWSN with large rmax might not exist even for |U | = 2 and unit weights. Two main types of weights are considered in the literature: the edge-weights and the node-weights. We consider the latter, which are usually more general than the former. For most undirected network design problems, a simple reduction transforms edge-weights to node-weights, but the inverse is usually not true. The study of network design problems with node-weights is well motivated and established from both theoretical as well as practical considerations; cf. [18,14,23,4,21]. For example, in telecommunication networks, expensive equipment, such as routers, switches, and transmitters, is located at the nodes of the network, and thus it is natural to model these problems by assigning weights to the nodes and/or to the edges, rather than to the edges only.In directed graphs, it is often possible to reduce the node-weights case to the edgeweights case via an approximation ratio preserving reduction. However, this is usually not so for undirected graphs, and an attempt to transform an undirected problem * Received by the editors July 9, 2008; accepted for publication (in revised form) March 23, 2010; published electronically June 9, 2010. A preliminary version of this paper appeared as [25].http://www.siam.org/journals/sicomp/39-7/72964.html † Department of Computer Science, The Open University of Israel, Raanana 43107, Israel (nutov@ openu.ac.il). 3001 3002ZEEV NUTOV into a directed one typically results in a problem which is significantly harder to approximate. For example, on undire...

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Approximating Minimum Cost Connectivity Problems via Uncrossable Bifamilies and Spider-Cover Decompositions

Cited by 31 publications

References 34 publications

A note on Rooted Survivable Networks

A note on Rooted Survivable Networks

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Approximating Steiner Networks with Node-Weights

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