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What Would Edmonds Do? Augmenting Paths and Witnesses for DegreeBounded MSTs
 IN PROCEEDINGS OF APPROX/RANDOM
, 2005
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Design and Optimization of a Tiered Wireless Access Network
"... Abstract—Although having high potential for broadband wireless access, wireless mesh networks are known to suffer from throughput and fairness problems, and are thus hard to scale to large size. To this end, hierarchical architectures provide a solution to this scalability problem. In this paper, w ..."
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Abstract—Although having high potential for broadband wireless access, wireless mesh networks are known to suffer from throughput and fairness problems, and are thus hard to scale to large size. To this end, hierarchical architectures provide a solution to this scalability problem. In this paper, we address the problem of design and optimization of a tiered wireless access network. At the lower tier, mesh routers are clustered based on traffic demands and delay requirements. The cluster heads are equipped with wireless optical transceivers and form the upper tier free space optical (FSO) network. We first present a plane sweeping and clustering algorithm aiming to minimize the number of clusters. PSC sweeps the network area and captures cluster members under delay and traffic load constraints. We then present an algebraic connectivitybased formulation for FSO network topology optimization and develop a greedy edgeappending algorithm that iteratively inserts edges to maximize algebraic connectivity. The proposed algorithms are analyzed and evaluated via simulations, and are shown to be highly effective as compared to the performance bounds derived in this paper. I.
Degree Bounded Network Design with Metric Costs
"... Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning ..."
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Cited by 8 (3 self)
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Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NPhard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for various degree bounded network design problems. • Global edgeconnectivity: There is a (2 + 1 k)approximation algorithm for the minimum bounded degree kedgeconnected subgraph problem. • Local edgeconnectivity: There is a 6approximation algorithm for the minimum bounded degree Steiner network problem. • Global vertexconnectivity: There is a (2 + k−1 n + 1 k)approximation algorithm for the minimum bounded degree kvertexconnected subgraph problem. • Spanning tree: There is an (1 + 1 d−1)approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with smallest possible maximum degree, and the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides’ algorithm for metric TSP. The main technical tool is a simplicitypreserving edge splittingoff operation, which is used to “shortcut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.
On the Design and Optimization of a Free Space Optical Access Network
"... Although having high potential for broadband wireless access, wireless mesh networks are known to suffer from throughput and fairness problems, and are thus hard to scale to large size. To this end, hierarchical architectures provide a solution to this scalability problem. In this paper, we address ..."
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Although having high potential for broadband wireless access, wireless mesh networks are known to suffer from throughput and fairness problems, and are thus hard to scale to large size. To this end, hierarchical architectures provide a solution to this scalability problem. In this paper, we address the problem of design and optimization of a tiered wireless access network that exploits free space optical (FSO) communications. The lower tier consists of mesh routers that are clustered based on traffic demands and delay requirements. The cluster heads are equipped with wireless optical transceivers and form the upper tier FSO network. For topology design and optimization, we first present a plane sweeping and clustering (PSC) algorithm aiming to minimize the total number of clusters. PSC sweeps the network area and captures cluster members under delay and traffic load constraints. For the upper tier FSO network, we present an algebraic connectivitybased formulation for topology optimization. We then develop a greedy edgeappending (GEA) algorithm, as well as its distributed version, that iteratively inserts edges to maximize algebraic connectivity. The proposed algorithms are analyzed and evaluated via simulations, and are shown to be highly effective as compared to the performance bounds derived in this paper.
Approximation Algorithms for the SingleSink Edge Installation Problems and Other Graph Problems
, 2004
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"... Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.561 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal de ..."
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Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.561 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal degree 3 exists that has this ratio be less than 1.447. Our central result is based on the proof of the following claim: Given n points in Euclidean space with one special point V, there exists a Hamiltonian path with an endpoint at V that is at most 1.561 times longer than the sum of the distances of the points to V. These proofs also lead to a way to find the tree in linear time given the minimal spanning tree. 2
BoundedAngle Spanning Tree: Modeling Networks with Angular Constraints∗
, 2014
"... We introduce a new structure for a set of points in the plane and an angle α, which is similar in flavor to a boundeddegree MST. We name this structure αMST. Let P be a set of points in the plane and let 0 < α ≤ 2pi be an angle. An αST of P is a spanning tree of the complete Euclidean graph in ..."
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We introduce a new structure for a set of points in the plane and an angle α, which is similar in flavor to a boundeddegree MST. We name this structure αMST. Let P be a set of points in the plane and let 0 < α ≤ 2pi be an angle. An αST of P is a spanning tree of the complete Euclidean graph induced by P, with the additional property that for each point p ∈ P, the smallest angle around p containing all the edges adjacent to p is at most α. An αMST of P is then an αST of P of minimum weight. For α < pi/3, an αST does not always exist, and, for α ≥ pi/3, it always exists [1, 2, 9]. In this paper, we study the problem of computing an αMST for several common values of α. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point p ∈ P, we associate a wedge wp of angle α and apex p. The goal is to assign an orientation and a radius rp to each wedge wp, such that the resulting graph is connected and its MST is an αMST. (We draw an edge between p and q if p ∈ wq, q ∈ wp, and pq  ≤ rp, rq.) Unsurprisingly, the problem of computing an αMST is NPhard, at least for α = pi and α = 2pi/3. We present constantfactor approximation algorithms for α = pi/2, 2pi/3, pi. One of our major results is a surprising theorem for α = 2pi/3, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set P of 3n points in the plane and any partitioning of the points into n triplets, one can orient the wedges of each triplet independently, such that the graph induced by P is connected. We apply the theorem to the antenna conversion problem. 1