Abstract
The local search framework for obtaining PTASs for NP-hard geometric optimization problems was introduced, independently, by Chan and Har-Peled [6] and Mustafa and Ray [17]. In this paper, we generalize the framework by extending its analysis to additional families of graphs, beyond the family of planar graphs. We then present several applications of the generalized framework, some of which are very different from those presented to date (using the original framework). These applications include PTASs for finding a maximum l-shallow set of a set of fat objects, for finding a maximum triangle matching in an l-shallow unit disk graph, and for vertex-guarding a (not-necessarily-simple) polygon under an appropriate shallowness assumption.
We also present a PTAS (using the original framework) for the important problem where one has to find a minimum-cardinality subset of a given set of disks (of varying radii) that covers a given set of points, and apply it to a class cover problem (studied in [3]) to obtain an improved solution.
Work by R. Aschner was partially supported by the Lynn and William Frankel Center for Computer Sciences. Work by R. Aschner and M.J. Katz was partially supported by the Israel Ministry of Industry, Trade and Labor (consortium CORNET). Work by M.J. Katz was partially supported by grant 1045/10 from the Israel Science Foundation, and by grant 2010074 from the United States – Israel Binational Science Foundation. Work by G. Morgenstern was partially supported by the Caesarea Rothschild Institute (CRI).
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Aschner, R., Katz, M.J., Morgenstern, G., Yuditsky, Y. (2013). Approximation Schemes for Covering and Packing. In: Ghosh, S.K., Tokuyama, T. (eds) WALCOM: Algorithms and Computation. WALCOM 2013. Lecture Notes in Computer Science, vol 7748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36065-7_10
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