Approximation algorithms for the connected sensor cover problem

Abstract We study the minimum connected sensor cover problem ( MIN - CSC ) and the budgeted connected sensor cover ( Budgeted - CSC ) problem, both motivated by important applications (e.g., reduce the communication cost among sensors) in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN - CSC , our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation algorithm, assuming that the ratio between the sensor radius and communication radius is bounded. In Budgeted - CSC problem, our goal is to choose a set of B sensors, such that the number of targets covered by the chosen sensors is maximized and the communication graph is connected. We also obtain a constant approximation under the same assumption.

[1]  Weili Wu,et al.  Connected Set-Cover and Group Steiner Tree , 2016, Encyclopedia of Algorithms.

[2]  Samir Khuller,et al.  Analyzing the Optimal Neighborhood: Algorithms for Budgeted and Partial Connected Dominating Set Problems , 2013, SODA.

[3]  Michael T. Goodrich,et al.  Almost optimal set covers in finite VC-dimension , 1995, Discret. Comput. Geom..

[4]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[5]  David Steurer,et al.  Analytical approach to parallel repetition , 2013, STOC.

[6]  Weili Wu,et al.  Minimum Connected Sensor Cover , 2016, Encyclopedia of Algorithms.

[7]  Chandra Chekuri,et al.  Submodular function maximization via the multilinear relaxation and contention resolution schemes , 2011, STOC '11.

[8]  Changyuan Yu,et al.  A 5+epsilon-approximation algorithm for minimum weighted dominating set in unit disk graph , 2009, Theor. Comput. Sci..

[9]  Michael Segal,et al.  Improved approximation algorithms for connected sensor cover , 2007, Wirel. Networks.

[10]  Kunal Talwar,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2004, J. Comput. Syst. Sci..

[11]  Weili Wu,et al.  A better constant-factor approximation for weighted dominating set in unit disk graph , 2009, J. Comb. Optim..

[12]  Harry B. Hunt,et al.  Simple heuristics for unit disk graphs , 1995, Networks.

[13]  Samir Khuller,et al.  Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets , 1998, Inf. Comput..

[14]  D. Hochbaum,et al.  Analysis of the greedy approach in problems of maximum k‐coverage , 1998 .

[15]  Samir R. Das,et al.  Connected sensor cover: self-organization of sensor networks for efficient query execution , 2006, TNET.

[16]  Gruia Calinescu,et al.  The Polymatroid Steiner Problems , 2005, J. Comb. Optim..

[17]  Kenneth L. Clarkson,et al.  Improved Approximation Algorithms for Geometric Set Cover , 2007, Discret. Comput. Geom..

[18]  R. Ravi,et al.  A polylogarithmic approximation algorithm for the group Steiner tree problem , 2000, SODA '98.

[19]  Nabil H. Mustafa,et al.  PTAS for geometric hitting set problems via local search , 2009, SCG '09.

[20]  Ion I. Mandoiu,et al.  Selecting Forwarding Neighbors in Wireless Ad Hoc Networks , 2004, Mob. Networks Appl..

[21]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[22]  Naveen Garg,et al.  Saving an epsilon: a 2-approximation for the k-MST problem in graphs , 2005, STOC '05.

[23]  Kasturi R. Varadarajan.  Weighted geometric set cover via quasi-uniform sampling , 2010, STOC '10.

[24]  Saurabh Ray,et al.  New existence proofs ε-nets , 2008, SCG '08.

[25]  C. Jordan.  Sur les assemblages de lignes. , 1869 .

[26]  Weili Wu,et al.  Complexity and approximation of the connected set-cover problem , 2011, Journal of Global Optimization.

[27]  Khaled M. Elbassioni,et al.  The relation of Connected Set Cover and Group Steiner Tree , 2012, Theor. Comput. Sci..

[28]  Kate Ching-Ju Lin,et al.  Maximizing Submodular Set Function With Connectivity Constraint: Theory and Application to Networks , 2015, IEEE/ACM Trans. Netw..

[29]  David P. Williamson,et al.  A general approximation technique for constrained forest problems , 1992, SODA '92.

[30]  Jan Vondrák,et al.  Maximizing a Monotone Submodular Function Subject to a Matroid Constraint , 2011, SIAM J. Comput..

[31]  Thomas Erlebach,et al.  Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs , 2006, APPROX-RANDOM.

[32]  Timothy M. Chan,et al.  Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling , 2012, SODA.

[33]  Jing Lv,et al.  Approximations for Minimum Connected Sensor Cover , 2013, 2013 Proceedings IEEE INFOCOM.

[34]  Jian Li,et al.  A PTAS for the Weighted Unit Disk Cover Problem , 2015, ICALP.

[35]  R. Ravi,et al.  A nearly best-possible approximation algorithm for node-weighted Steiner trees , 1993, IPCO.

[36]  Guy Kortsarz,et al.  A greedy approximation algorithm for the group Steiner problem , 2006, Discret. Appl. Math..

[37]  Weili Wu,et al.  New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit disk graphs , 2011, Theor. Comput. Sci..

[38]  David S. Johnson,et al.  The prize collecting Steiner tree problem: theory and practice , 2000, SODA '00.

[39]  Harry B. Hunt,et al.  NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs , 1998, J. Algorithms.

[40]  Dror Rawitz,et al.  Hitting sets when the VC-dimension is small , 2005, Inf. Process. Lett..

[41]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[42]  Peng-Jun Wan,et al.  Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[43]  Deying Li,et al.  A polynomial‐time approximation scheme for the minimum‐connected dominating set in ad hoc wireless networks , 2003, Networks.