Maybe there could be some other encodings in which only unit disk graphs could be represented, but formulating them would be another topic.)Įdit: I'll try to formalize the question and the answer more: (Here we assume that the input is some well-known encoding of a graph, and therefore the restriction that the input must be an unit disk graph doesn't really make the problem at all easier. Therefore the problem you pose is NP-hard in the sense that if it admits polynomial time algorithm, then P=NP. Vazirani, V.V.: Approximation Algorithms.If there would be a polynomial time algorithm for your problem, it could be used to solve the NP-hard recognition problem in polynomial time by just giving the input to it and checking if its output is correct. Varadarajan, K.: Weighted geometric set cover via quasi-uniform sampling. (eds.) Approximation, Randomization, and Combinatorial Optimization. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. Pandit, S., Pemmaraju, S., Varadarajan, K.: Approximation algorithms for domatic partition. Nieberg, T., Hurink, J., Kern, W.: Approximation schemes for wireless networks. Mustafa, N.H., Ray, S.: PTAS for geometric hitting set problems via local search. Marx, D.: On the optimality of planar and geometric approximation schemes. Kammer, F., Tholey, T.: Approximation algorithms for intersection graphs. Impagliazzo, R., Paturi, R.: On the complexity of k-sat. 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. Gibson, M., Pirwani, I.A.: Approximation algorithms for dominating set in disk graphs. Springer, Heidelberg (2008)įeige, U.: A threshold of ln n for approximating set cover. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. Comput. 34(6), 1302–1323 (2005)Įrlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs. 1–11 (1988)Įrlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. In: Symposium on Computational Geometry, pp. Discrete Mathematics 86(1-3), 165–177 (1990)Ĭlarkson, K.L.: Applications of random sampling in computational geometry, II. Springer, Heidelberg (2004)Ĭlark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. 333–340 (2009)Ĭhlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems. Algorithms 46(2), 178–189 (2003)Ĭhan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Surv. 23(3), 345–405 (1991)Ĭhan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. Springer, Heidelberg (2006)Īurenhammer, F.: Voronoi diagrams-a survey of a fundamental geometric data structure. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. This process is experimental and the keywords may be updated as the learning algorithm improves.Īmbühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. These keywords were added by machine and not by the authors. We improve the status quo in two ways: for the unweighted case, we show how to obtain a PTAS using the framework recently proposed (independently) by Mustafa and Ray and by Chan and Har-Peled for the weighted case where each input disk has an associated rational weight with the objective of finding a minimum cost dominating set, we give a randomized algorithm that obtains a dominating set whose weight is within a factor \(2^\) of a minimum cost solution, with high probability – the technique follows the framework proposed recently by Varadarajan. The problem has been extensively studied on subclasses of disk graphs, yet the best known approximation for disk graphs has remained O(log n) – a bound that is asymptotically no better than the general case. We consider the problem of finding a lowest cost dominating set in a given disk graph containing n disks.
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