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92
Algorithmic Aspects of Topology Control Problems for Ad hoc Networks
, 2002
"... Topology control problems are concerned with the assignment of power values to the nodes of an ad~hoc network so that the power assignment leads to a graph topology satisfying some specified properties. This paper considers such problems under several optimization objectives, including minimizing th ..."
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Cited by 119 (6 self)
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Topology control problems are concerned with the assignment of power values to the nodes of an ad~hoc network so that the power assignment leads to a graph topology satisfying some specified properties. This paper considers such problems under several optimization objectives, including minimizing the maximum power and minimizing the total power. A general approach leading to a polynomial algorithm is presented for minimizing maximum power for a class of graph properties called monotone properties. The difficulty of generalizing the approach to properties that are not monotone is discussed. Problems involving the minimization of total power are known to be NPcomplete even for simple graph properties. A general approach that leads to an approximation algorithm for minimizing the total power for some monotone properties is presented. Using this approach, a new approximation algorithm for the problem of minimizing the total power for obtaining a 2nodeconnected graph is obtained. It is shown that this algorithm provides a constant performance guarantee. Experimental results from an implementation of the approximation algorithm are also presented.
RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS
, 1999
"... We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for pro ..."
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Cited by 92 (11 self)
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We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cutapproximation algorithms extend unchanged to weighted graphs while our weightedgraph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a nearlinear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cutbased problems, including approximating the best balanced cut of a graph, finding a kconnected orientation of a 2kconnected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum kconnected subgraph problem from 1.85 to 1 � O(�log n)/k).
A primaldual approximation algorithm for generalized Steiner network problems
 Combinatorica
, 1995
"... M.I.T. We present the first polynomialtime approximation algorithm for finding a minimumcost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. ..."
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Cited by 90 (18 self)
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M.I.T. We present the first polynomialtime approximation algorithm for finding a minimumcost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. If k is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primaldual and shows the importance of this technique in designing approximation algorithms. 1
Power Optimization in FaultTolerant Topology Control Algorithms for Wireless Multihop Networks
 in Proceedings of the 9th Annual International Conference on Mobile Computing and Networking. 2003
, 2003
"... In ad hoc wireless networks, it is crucial to minimize power consumption while maintaining key network properties. This work studies power assignments of wireless devices that minimize power while maintaining kfault tolerance. Specifically, we require all links established by this power setting be ..."
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Cited by 81 (6 self)
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In ad hoc wireless networks, it is crucial to minimize power consumption while maintaining key network properties. This work studies power assignments of wireless devices that minimize power while maintaining kfault tolerance. Specifically, we require all links established by this power setting be symmetric and form a kvertex connected subgraph of the network graph. This problem is known to be NPhard. We show current heuristic approaches can use arbitrarily more power than the optimal solution. Hence, we seek approximation algorithms for this problem. We present three approximation algorithms. The first algorithm gives an O(kα)approximation where α is the best approximation factor for the related problem in wired networks (the best α so far is O(log k).) With a more careful analysis, we show our second (slightly more complicated) algorithm is an O(k)approximation. Our third algorithm assumes that the edge lengths of the network graph form a metric. In this case, we present simple and practical distributed algorithms for the cases of 2 and 3connectivity with constant approximation factors. We generalize this algorithm to obtain an O(k 2c+2)approximation for general kconnectivity (2 ≤ c ≤ 4 is the power attenuation exponent). Finally, we show that these approximation algorithms compare favorably with existing heuristics. We note that all algorithms presented in this paper can be used to minimize power while maintaining kedge connectivity with guaranteed approximation factors.
Improved Approximation Algorithms for Uniform Connectivity Problems
 J. Algorithms
"... The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NPhard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. Th ..."
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Cited by 78 (3 self)
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The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NPhard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. The following results are presented: 1. For the unweighted kedgeconnectivity problem an approximation algorithm that achieves a performance ratio of 1.85 is described. This is the first polynomialtime algorithm that achieves a constant less than 2, for all k. 2. For the weighted kvertexconnectivity problem, a constant factor approximation algorithm is given assuming that the edgeweights satisfy the triangle inequality. This is the first constant factor approximation algorithm for this problem. 3. For the case of biconnectivity, with no assumptions about the weights of the edges, an algorithm that achieves a factor asymptotically approaching 2 is described. This matches the previous best...
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
"... ..."
Approximation Algorithms for Finding Highly Connected Subgraphs
, 1996
"... Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 EdgeConnectivity Problems 3 2.1 Weighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted EdgeConnectivity : : : : : ..."
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Cited by 70 (1 self)
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Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 EdgeConnectivity Problems 3 2.1 Weighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 VertexConnectivity Problems 11 3.1 Weighted VertexConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted VertexConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 VertexConnectivity : : : : : : : : : : : : : : : : :
An Efficient Approximation Algorithm for the Survivable Network Design Problem
 IN PROCEEDINGS OF THE THIRD MPS CONFERENCE ON INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION
, 1993
"... The survivable network design problem is to construct a minimumcost subgraph satisfying certain given edgeconnectivity requirements. The first polynomialtime approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph ..."
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Cited by 54 (6 self)
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The survivable network design problem is to construct a minimumcost subgraph satisfying certain given edgeconnectivity requirements. The first polynomialtime approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph
APPROXIMATING THE MINIMUM EQUIVALENT DIGRAPH
, 1995
"... The minimum equivalent graph (MEG) problem is as follows: given a directed graph, find a smallest subset of the edges that maintains all teachability relations between nodes. This problem is NPhard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in poly ..."
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Cited by 42 (2 self)
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The minimum equivalent graph (MEG) problem is as follows: given a directed graph, find a smallest subset of the edges that maintains all teachability relations between nodes. This problem is NPhard; this paper gives an approximation algorithm achieving a performance guarantee of about 1.64 in polynomial time. The algorithm achieves a performance guarantee of 1.75 in the time required for transitive closure. The heart of the MEG problem is the minimum strongly connected spanning subgraph (SCSS) problemthe MEG problem restricted to strongly connected digraphs. For the minimum SCSS problem, the paper gives apractical, nearly lineartime implementation achieving a performance guarantee of 1.75. The algorithm and its analysis are based on the simple idea of contracting long cycles. The analysis applies directly to2EXCHANCE, a general "local improvement" algorithm, showing that its performance guarantee is 1.75.