Abstract
We consider systems of small, cheap, simple sensors that are organized in a distributed network and used for estimating and tracking the locations of targets. The objective is to assign sensors to targets such that the overall expected error of the sensors’ estimates of the target locations is minimized. The so-called focus of attention problem (FOA) deals with the special case where every target is tracked by one pair of range sensors. The resulting computational problem is a special case of the axial three-index assignment problem, a well-known fundamental problem in combinatorial optimization. We provide a complete complexity and approximability analysis of the FOA problem: we establish strong NP-hardness and the unlikeliness of an FPTAS, we identify polynomially solvable special cases, and we construct a sophisticated polynomial time approximation scheme for it.
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Al-Hasan, M., Ramachandran, K.K., Mitchell, J.E.: Optimal placement of stereo sensors. Optim. Lett. 2, 99–111 (2008)
Burkard, R.E., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009)
Burkard, R.E., Rudolf, R., Woeginger, G.J.: Three-dimensional axial assignment problems with decomposable cost coefficients. Discret. Appl. Math. 65, 123–139 (1996)
Crama, Y., Spieksma, F.C.R.: Approximation algorithms for three-dimensional assignment problems with triangle inequalities. Eur. J. Oper. Res. 60, 273–279 (1992)
Culler, D., Estrin, D., Srivastava, M.: Guest editors’ introduction: overview of sensor networks. IEEE Comput. 37, 41–49 (2004)
Gfeller, B., Mihalak, M., Suri, S., Vicari, E., Widmayer, P.: Angle optimization in target tracking. In: Proceedings of the 11th Scandinavian Workshop on Algorithm Theory (SWAT’2008), Springer LNCS, vol. 5124, pp. 65–76 (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Isler, V., Khanna, S., Spletzer, J.R., Taylor, C.J.: Target tracking with distributed sensors: the focus of attention problem. Comput. Vis. Image Underst. 100, 225–247 (2005)
Queyranne, M., Spieksma, F.C.R.: Approximation algorithms for multi-index transportation problems with decomposable costs. Discret. Appl. Math. 76, 239–253 (1997)
Spieksma, F.C.R., Woeginger, G.J.: Geometric three-dimensional assignment problems. Eur. J. Oper. Res. 91, 611–618 (1996)
Tubaishat, M., Madria, S.: Sensor networks: an overview. IEEE Potentials 22, 20–23 (2003)
Woeginger, G.J.: When does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)? INFORMS J. Comput. 12, 57–74 (2000)
Yu, W., Hoogeveen, H., Lenstra, J.K.: Minimizing makespan in a two-machine flow shop with delays and unit-time operations is NP-hard. J. Sched. 7, 333–348 (2004)
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A preliminary version of this work has appeared in the Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), 312–317. This research has been supported by the Netherlands Organisation for Scientific Research (NWO) under Grant 639.033.403, by BSIK Grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society), and by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.
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Goossens, D., Polyakovskiy, S., Spieksma, F.C.R. et al. The Focus of Attention Problem. Algorithmica 74, 559–573 (2016). https://doi.org/10.1007/s00453-014-9963-8
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DOI: https://doi.org/10.1007/s00453-014-9963-8