Approximation algorithms for finding low‐degree subgraphs

Klein, Philip N.; Krishnan, Radha; Raghavachari, Balaji; Ravi, R.

doi:10.1002/net.20031

Cited by 22 publications

(15 citation statements)

References 19 publications

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“…(Their result holds for the Steiner tree problem as well.) Ravi, Raghavachari, and Klein [36], and Klein et al [24] considered the Minimum Degree k-Edge-Connected Subgraph problem and gave an approximation algorithm with performance ratio O(n δ ) for any fixed δ > 0 in polynomial time and O(log n/ log log n) in subexponential time. Recently, Feder, Motwani, and Zhu [11] obtained a polynomial time O(k log n)-approximation algorithm for this problem for any fixed k, thus answering an open question in [36].…”

Section: Introductionmentioning

confidence: 99%

Survivable Network Design with Degree or Order Constraints

Lau¹,

Naor²,

Salavatipour³

et al. 2009

SIAM J. Comput.

101

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Abstract. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv +3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv + 3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

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Section: Introductionmentioning

confidence: 99%

Survivable Network Design with Degree or Order Constraints

Lau¹,

Naor²,

Salavatipour³

et al. 2009

SIAM J. Comput.

101

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“…In [22], Wang and Li present a localized algorithm for constructing bounded degree planar spanners, but the degree bound is high (20). In [23], Li, Hou, and Sha describe a minimum spanning tree based algorithm that builds connected subgraphs with a bound of 6 on node degree.…”

Section: B Bounded Degree Subgraphsmentioning

confidence: 98%

“…The problem of finding minimum degree two-connected spanning subgraphs is more challenging. To this end, Klein et al [20] provide a quasi-polynomial time approximation algorithm for finding two-edge-connected spanning subgraphs, whose degree in all cases is guaranteed to be at most 1 + times the optimal degree ∆ * , plus an additive O(log 1+ n) for any > 0. Thus, for large full graphs they cannot guarantee to find a two-connected subgraph with a small constant degree.…”

Section: B Bounded Degree Subgraphsmentioning

confidence: 98%

Building Robust Nomadic Wireless Mesh Networks Using Directional Antennas

Dong¹,

Bejerano²

2008

2008 Proceedings IEEE INFOCOM - The 27th Conference on Computer Communications

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Recently, wireless mesh technology has been used for military applications and fast recovery networks, referred to as nomadic wireless mesh networks (NWMNs). In such systems, wireless routers, termed nodes, are mounted on top of vehicles or vessels, which may change their location according to application needs; and the nodes are required to establish a reliable wireless mesh network. For improving network performance, some vendors use directional antennas, and the mesh topology comprises of point-to-point connections between adjacent nodes. The number of point-to-point connections of a node is upper-bounded by the number of directional radios it has, which is typically a small constant. This raises the need to build robust (i.e., two-node/edgeconnected) mesh networks with bounded node degree, regardless of node locations. In this paper, we present simple elegant schemes for constructing such efficient and robust wireless mesh networks with provably small constant degree bounds. Our extensive simulations show our schemes build robust and efficient topologies for various settings with node degree bounded by 4 and small hop-count distance between nodes and gateways.

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“…For bounded-degree spanning trees and Steiner trees, Fürer and Raghavachari [7] gave an approximation algorithm which violates the degrees by at most an additive constant one. This result has generated much interest in obtaining approximation algorithms for network design problems with degree constraints [12,13,11,14,6,18,4,5,19,20,8,21]. A highlight of this line of research is an (1, b v + 2)-approximation algorithm 1 for the minimum bounded degree spanning tree problem by Goemans [8].…”

Section: Related Workmentioning

confidence: 99%

Additive Approximation for Bounded Degree Survivable Network Design

Lau¹,

Singh²

2013

SIAM J. Comput.

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In the minimum bounded degree Steiner network problem, we are given an undirected graph with an edge cost for each edge, a connectivity requirement r uv for each pair of vertices u and v, and a degree upper bound b v for each vertex v. The task is to find a minimum cost subgraph that satisfies all the connectivity requirements and degree upper bounds. Let r max := max u,v {r uv } and OPT be the cost of an optimal solution that satisfies all the degree bounds. We present approximation algorithms that minimize the total cost and the degree violation simultaneously.• In the special case when r max = 1, there is a polynomial time algorithm that returns a Steiner forest of cost at most 2OPT and the degree of each vertex v is at most b v + 3.• In the general case, there is a polynomial time algorithm that returns a Steiner network of cost at most 2OPT and the degree of each vertex v is at most b v + 6r max + 3.The algorithms are based on the iterative relaxation method, and the analysis of the algorithms is nearly tight. * A preliminary version appeared in

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Approximation algorithms for finding low‐degree subgraphs

Cited by 22 publications

References 19 publications

Survivable Network Design with Degree or Order Constraints

Survivable Network Design with Degree or Order Constraints

Building Robust Nomadic Wireless Mesh Networks Using Directional Antennas

Additive Approximation for Bounded Degree Survivable Network Design

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