Abstract
We study optimization problems for the Euclidean minimum spanning tree (MST) on imprecise data. To model imprecision, we accept a set of disjoint disks in the plane as input. From each member of the set, one point must be selected, and the MST is computed over the set of selected points. We consider both minimizing and maximizing the weight of the MST over the input. The minimum weight version of the problem is known as the minimum spanning tree with neighborhoods (\(\textsc{MSTN}\)) problem, and the maximum weight version (\(\textsc{max-MSTN}\)) has not been studied previously to our knowledge. We provide deterministic and parameterized approximation algorithms for the \(\textsc{max-MSTN}\) problem, and a parameterized algorithm for the \(\textsc{MSTN}\) problem. Additionally, we present hardness of approximation proofs for both settings.
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References
Arkin, E., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics 55(3), 197–218 (1994)
de Berg, M., Gudmundsson, J., Katz, M., Levcopoulos, C., Overmars, M., van der Stappen, A.: TSP with neighborhoods of varying size. Journal of Algorithms 57(1), 22–36 (2005)
Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM 42(1), 67–90 (1995)
Dorrigiv, R., Fraser, R., He, M., Kamali, S., Kawamura, A., López-Ortiz, A., Seco, D.: On minimum- and maximum-weight minimum spanning trees with neighborhoods. Tech. Rep. CS-2012-14, University of Waterloo (2012)
Dumitrescu, A., Mitchell, J.S.: Approximation algorithms for TSP with neighborhoods in the plane. Journal of Algorithms 48(1), 135–159 (2003)
Erlebach, T., Hoffmann, M., Krizanc, D., Mihalák, M., Raman, R.: Computing minimum spanning trees with uncertainty. In: Symposium on Theoretical Aspects of Computer Science, pp. 277–288 (2008)
Fiala, J., Kratochvíl, J., Proskurowski, A.: Systems of distant representatives. Discrete Applied Mathematics 145(2), 306–316 (2005)
Graham, R.L., Hell, P.: On the history of the minimum spanning tree problem. IEEE Annals of the History of Computing 7(1), 43–57 (1985)
Lichtenstein, D.: Planar formulae and their uses. SIAM J. on Computing 11(2), 329–344 (1982)
Löffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica 56, 235–269 (2010)
Nešetřil, J., Milková, E., Nešetřilová, H.: Otakar Borůvka on minimum spanning tree problem translation of both the 1926 papers, comments, history. Discrete Mathematics 233(13), 3–36 (2001)
Yang, Y.: On several geometric network design problems. Ph.D. thesis, State University of New York at Buffalo (2008)
Yang, Y., Lin, M., Xu, J., Xie, Y.: Minimum spanning tree with neighborhoods. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 306–316. Springer, Heidelberg (2007)
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Dorrigiv, R. et al. (2013). On Minimum-and Maximum-Weight Minimum Spanning Trees with Neighborhoods. In: Erlebach, T., Persiano, G. (eds) Approximation and Online Algorithms. WAOA 2012. Lecture Notes in Computer Science, vol 7846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38016-7_9
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DOI: https://doi.org/10.1007/978-3-642-38016-7_9
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