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81
Approximation Algorithms for Connected Dominating Sets
 Algorithmica
, 1996
"... The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, whe ..."
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Cited by 277 (9 self)
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The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...
When trees collide: An approximation algorithm for the generalized Steiner problem on networks
, 1994
"... We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the a ..."
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Cited by 219 (32 self)
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We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimumcost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 147 (12 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Approximation Algorithms for Directed Steiner Problems
 Journal of Algorithms
, 1998
"... We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work we ..."
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Cited by 143 (8 self)
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We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i \Gamma 1)k 1=i in time O(n i k 2i ) for any fixed i ? 1, where k is the number of terminals. Thus, an O(k ffl ) approximation ratio can be achieved in polynomial time for any fixed ffl ? 0. Setting i = log k, we obtain an O(log 2 k) approximation ratio in quasipolynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k 2=3 log 1=3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner...
A polylogarithmic approximation algorithm for the group Steiner tree problem
 Journal of Algorithms
, 2000
"... The group Steiner tree problem is a generalization of the Steiner tree problem where we ae given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimumweight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich a ..."
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Cited by 134 (9 self)
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The group Steiner tree problem is a generalization of the Steiner tree problem where we ae given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimumweight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich and Widmayer and finds applications in VLSI design. The group Steiner tree problem generalizes the set covering problem, and is therefore at least as had. We give a randomized O(log 3 n log k)approximation algorithm for the group Steiner tree problem on an nnode graph, where k is the number of groups. The best previous ink)v/ (Bateman, Helvig, performance guarantee was (1 +  Robins and Zelikovsky).
Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets
 Information and Computation
, 1999
"... A greedy approximation algorithm based on \spider decompositions " was developed by Klein and Ravi for node weighted Steiner trees. This algorithm provides a worst case approximation ratio of 2 ln k, where k is the number of terminals. However, the best known lower bound on the approximation ra ..."
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Cited by 68 (1 self)
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A greedy approximation algorithm based on \spider decompositions " was developed by Klein and Ravi for node weighted Steiner trees. This algorithm provides a worst case approximation ratio of 2 ln k, where k is the number of terminals. However, the best known lower bound on the approximation ratio is ln k, assuming that NP 6 DT IM E[n O(log log n)], by a reduction from set cover [9, 4]. We show that for the unweighted case we can obtain an approximation factor of ln k. For the weighted case we develop a new decomposition theorem, and generalize the notion of \spiders " to \branchspiders", that are used to design a new algorithm with a worst case approximation factor of 1:5lnk. This algorithm, although polynomial, is not very practical due to its high running time; since we need to repeatedly nd many minimum weight matchings in each iteration. We are able to generalize the method to yield an approximation factor approaching 1:35 ln k. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1:6103 ln k. The techniques developed for the second algorithm imply a method of approximating node weighted network design problems de ned by 01 proper functions. These new ideas also lead to improved approximation guarantees for the problem of nding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n [7]. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor approaching 1:35 ln n. 1.
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
"... ..."
Approximation algorithms for nonuniform buyatbulk network design problems
 Proc. of IEEE FOCS
"... Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorith ..."
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Cited by 58 (15 self)
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Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buyatbulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first polylogarithmic approximation ratio for the nonuniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Polylogarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems. Key words. Nonuniform buyatbulk, network design, approximation algorithm, concave cost, network flow, economies of scale AMS subject classifications. 68Q25, 68W25, 90C27, 90C59 1. Introduction. Network
Network Lifetime and Power Assignment in AdHoc Wireless Networks
 in ESA
, 2003
"... Abstract. Used for topology control in adhoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity) The input consists of a directed complete weighted graph G = (V; c). The power of a vertex u in a directed spanni ..."
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Cited by 42 (3 self)
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Abstract. Used for topology control in adhoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity) The input consists of a directed complete weighted graph G = (V; c). The power of a vertex u in a directed spanning subgraph H is given by pH(u) = maxuv2E(H) c(uv). The power of H is given by p(H) = P u2V pH(u), Power Assignment seeks to minimize p(H) while H satisfies the given connectivity constraint. We
A series of approximation algorithms for the Acyclic Directed Steiner Tree problem
 Algorithmica
, 1997
"... Abstract Given an acyclic directed network, a subset S of nodes (terminals), and a root r, the acyclic directed Steiner tree problem requires a minimumcost subnetwork which contains paths from r to each terminal. It is known that unless NP ` DT IME[npolylogn] no polynomialtime algorithm can guaran ..."
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Cited by 38 (1 self)
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Abstract Given an acyclic directed network, a subset S of nodes (terminals), and a root r, the acyclic directed Steiner tree problem requires a minimumcost subnetwork which contains paths from r to each terminal. It is known that unless NP ` DT IME[npolylogn] no polynomialtime algorithm can guarantee better then (ln k)=4 approximation, where k is the number of terminals. In this paper we give an O(kffl)approximation algorithm for any ffl? 0. This result improves the previously known kapproximation.