Abstract
We give a polynomial-time approximation scheme for the unique unit-square coverage problem: given a set of points and a set of axis-parallel unit squares, both in the plane, we wish to find a subset of squares that maximizes the number of points contained in exactly one square in the subset. Erlebach and van Leeuwen (2008) introduced this problem as the geometric version of the unique coverage problem, and the best approximation ratio by van Leeuwen (2009) before our work was 2. Our scheme can be generalized to the budgeted unique unit-square coverage problem, in which each point has a profit, each square has a cost, and we wish to maximize the total profit of the uniquely covered points under the condition that the total cost is at most a given bound.
This work is partially supported by Grant-in-Aid for Scientific Research, and by the Funding Program for World-Leading Innovative R&D on Science and Technology, Japan.
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References
Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)
Bazgan, C.: Schémas d’approximation et complexité paramétrée. Rapport de DEA, Université Paris Sud (1995)
Cesati, M., Trevisan, L.: On the efficiency of polynomial time approximation schemes. Information Processing Letters 64, 165–171 (1997)
Chandra, A.K., Hirschberg, D.S., Wong, C.K.: Approximate algorithms for some generalized knapsack problems. Theoretical Computer Science 3, 293–304 (1976)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86, 165–177 (1990)
Demaine, E.D., Hajiaghayi, M.T., Feige, U., Salavatipour, M.R.: Combination can be hard: approximability of the unique coverage problem. SIAM J. on Computing 38, 1464–1483 (2008)
Dumitrescu, A., Jiang, M.: Dispersion in unit disks. In: Proc. STACS 2010, pp. 299–310 (2010)
Erlebach, T., van Leeuwen, E.J.: Approximating geometric coverage problems. In: Proc. SODA 2008, pp. 1267–1276 (2008)
Fiala, J., Kratochvíl, J., Proskurowski, A.: Systems of distant representatives. Discrete Applied Mathematics 145, 306–316 (2005)
Guruswami, V., Trevisan, L.: The Complexity of Making Unique Choices: Approximating 1-in-k SAT. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 99–110. Springer, Heidelberg (2005)
Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)
Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26, 238–274 (1998)
Huson, M.L., Sen, A.: Broadcast scheduling algorithms for radio networks. In: Proc. IEEE MILCOM 1995, pp. 647–651 (1995)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer (2004)
Misra, N., Raman, V., Saurabh, S., Sikdar, S.: The Budgeted Unique Coverage Problem and Color-Coding. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) CSR 2009. LNCS, vol. 5675, pp. 310–321. Springer, Heidelberg (2009)
Moser, H., Raman, V., Sikdar, S.: The Parameterized Complexity of the Unique Coverage Problem. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 621–631. Springer, Heidelberg (2007)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)
van Leeuwen, E.J.: Optimization and approximation on systems of geometric objects. Ph.D. Thesis, University of Amsterdam (2009)
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Ito, T. et al. (2012). A Polynomial-Time Approximation Scheme for the Geometric Unique Coverage Problem on Unit Squares. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_3
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