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21
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 ...
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...
Connected Sensor Cover: SelfOrganization of Sensor Networks for Efficient Query Execution
 MOBIHOC'03
, 2003
"... Spatial query execution is an essential functionality of a sensor network, where a query gathers sensor data within a specific geographic region. Redundancy within a sensor network can be exploited to rv uce the communication cost incurv1 in execution of such quer ies. Anyr eduction in communicatio ..."
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Cited by 107 (6 self)
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Spatial query execution is an essential functionality of a sensor network, where a query gathers sensor data within a specific geographic region. Redundancy within a sensor network can be exploited to rv uce the communication cost incurv1 in execution of such quer ies. Anyr eduction in communication cost wouldr esult in an e#cient use of the batter y ener gy, which is ver y limited in sensor s. One appr oach to r educe the communication cost of a quer y is to selfor ganize the networ# inr esponse to a quer , into a topology that involves only a small subset of the sensor s su#cient to pr ocess the quer y. The quer y is then executed using only the sensor in the constr ucted topology. In thisar icle, we design and analyze algor thms for such selfor"/0 zation of asensor networ tor educe enerP consumption. In par icular we develop the notion of a connected sensor cover and design a centr alized appr oximation algor thm that constr ucts a topology in ol ing anear optimal connected sensor co er . We pr o e that the size of the const rst ed topology is within an O(log n)factor ofthe optimal size, wher n is the networ size. We also de elop a distr ibuted selfor$1" zationer" on ofour algor thm, and prv ose seer/ optimizations tor educe the communication oer"E1 of the algorithm. Finally, we evaluate the distributed algorithm using simulations and show that our approach results in significant communication cost reduction.
Biconnectivity Approximations and Graph Carvings
, 1994
"... A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be ..."
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Cited by 84 (3 self)
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A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP hard. We consider the problem of finding a better approximation to the smallest 2connected subgraph, by an efficient algorithm. For 2edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP hard as well. We also consider the case where the graph has edge weigh...
Approximation Algorithms For Geometric Problems
, 1995
"... INTRODUCTION 8.1 This chapter surveys approximation algorithms for hard geometric problems. The problems we consider typically take inputs that are point sets or polytopes in two or threedimensional space, and seek optimal constructions, (which may be trees, paths, or polytopes). We limit attent ..."
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Cited by 82 (1 self)
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INTRODUCTION 8.1 This chapter surveys approximation algorithms for hard geometric problems. The problems we consider typically take inputs that are point sets or polytopes in two or threedimensional space, and seek optimal constructions, (which may be trees, paths, or polytopes). We limit attention to problems for which no polynomialtime exact algorithms are known, and concentrate on bounds for worstcase approximation ratios, especially bounds that depend intrinsically on geometry. We illustrate our intentions with two wellknown problems. Given a finite set of points S in the plane, the Euclidean traveling salesman problem asks for the shortest tour of S. Christofides' algorithm achieves approximation ratio 3 2 for this problem, meaning that it always computes a tour of length at most threehalves the length of the optimal tour. This bound depends only on the triangle inequality, so Christofides' algorit
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.
Better approximation bounds for the network and Euclidean Steiner tree problems
, 1995
"... The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G = (V; E; d) and Euclidean plane, respectively. For these problems, we present a series of heuristics finding approximate Steiner tree with performance guarantee coming arbitrary ..."
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Cited by 43 (3 self)
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The network and Euclidean Steiner tree problems require a shortest tree spanning a given vertex subset within a network G = (V; E; d) and Euclidean plane, respectively. For these problems, we present a series of heuristics finding approximate Steiner tree with performance guarantee coming arbitrary close to 1+ln2 1:693 and 1+ln 2 p 3 1:1438, respectively. The best previously known corresponding values are close to 1.746 and 1.1546. Keywords: Combinatorial problems, approximation algorithms, Steiner trees. 1 Introduction Let G = (V; E;d) be a graph with a vertex set V , an edge set E and distance function d : E ! R + . A tree T is called a Steiner tree of S, S ae V , if S is contained in the vertex set of T . Network Steiner Problem (NSP). Given G and S, find the shortest Steiner tree (also called the Steiner minimal tree) of S. This problem is NPcomplete [9], so many approximation algorithms for Steiner minimal trees appeared in the last two decades. The quality of an appr...
New Approximation Algorithms for the Steiner Tree Problems
 Journal of Combinatorial Optimization
, 1995
"... The Steiner tree problem asks for the shortest tree connecting a given set of terminal points in a metric space. We design new approximation algorithms for the Steiner tree problems using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solutions. ..."
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Cited by 42 (5 self)
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The Steiner tree problem asks for the shortest tree connecting a given set of terminal points in a metric space. We design new approximation algorithms for the Steiner tree problems using a novel technique of choosing Steiner points in dependence on the possible deviation from the optimal solutions. We achieve the best up to now approximation ratios of 1.644 in arbitrary metric and 1.267 in rectilinear plane, respectively. Dept. of Computer Science, University of Bonn, 53117 Bonn, and International Computer Science Institute, Berkeley, California. Supported in part by the Leibniz Center for Research in Computer Science, by DFG Grant KA 673/41, by the ESPRIT BR Grants 7097 and by ECUS030. Email: marek@cs.unibonn.de. y Institute of Mathematics, 277028 Kishinev, Moldova. Research partially supported by Volkswagen Stiftung. Parts of this work were done in Max Planck Institute fur Informatik, Saarbrucken. Email: 17azz@mathem.moldova.su. 1 Introduction We consider a metric space wi...
Efficient gathering of correlated data in sensor networks
 in Proc. of ACM Intl. symposium on Mobile ad hoc networking and computing, 2005
, 2005
"... In this paper, we design techniques that exploit data correlations in sensor data to minimize communication costs (and hence, energy costs) incurred during data gathering in a sensor network. Our proposed approach is to select a small subset of sensor nodes that may be sufficient to reconstruct data ..."
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Cited by 38 (0 self)
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In this paper, we design techniques that exploit data correlations in sensor data to minimize communication costs (and hence, energy costs) incurred during data gathering in a sensor network. Our proposed approach is to select a small subset of sensor nodes that may be sufficient to reconstruct data for the entire sensor network. Then, during data gathering only the selected sensors need to be involved in communication. The selected set of sensors must also be connected, since they need to relay data to the datagathering node. We define the problem of selecting such a set of sensors as the connected correlationdominating set problem, and formulate it in terms of an appropriately defined correlation structure that captures general data correlations in a sensor network. We develop a set of energyefficient distributed algorithms and competitive centralized heuristics to select a connected correlationdominating set of small size. The designed distributed algorithms can be implemented in an asynchronous communication model, and can tolerate message losses. We also design an exponential (but nonexhaustive) centralized approximation algorithm that returns a solution within O(log n) of the optimal size. Based on the approximation algorithm, we design a class of centralized heuristics that are empirically shown to return nearoptimal solutions. Simulation results over randomly generated sensor networks with both artificially and naturally generated data sets demonstrate the efficiency of the designed algorithms and the viability of our technique – even in dynamic conditions.