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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

In this work, we address the following question. Suppose we are given a set D of positive-weighted disks and a set T of n points in the plane, such that each point of T is contained in at least two disks of D. Then is there always a subset S of D such that the union of the disks in S contains all the points of T and the total weight of the disks of D that are not in S is at least a constant fraction of the total weight of the disks in D?
In our work, we prove the Extraction Theorem that answers this question in the affirmative. Our constructive proof heavily exploits the geometry of disks, and in the process, we make interesting connections between our work and the literature on local search for geometric optimization problems.
The Extraction Theorem helps to design the first polynomial-time O(1)-approximations for two important geometric covering problems involving disks.

Sayan Bandyapadhyay, Anil Maheshwari, Sasanka Roy, Michiel Smid, and Kasturi Varadarajan. Geometric Covering via Extraction Theorem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ITCS.2024.7, author = {Bandyapadhyay, Sayan and Maheshwari, Anil and Roy, Sasanka and Smid, Michiel and Varadarajan, Kasturi}, title = {{Geometric Covering via Extraction Theorem}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.7}, URN = {urn:nbn:de:0030-drops-195355}, doi = {10.4230/LIPIcs.ITCS.2024.7}, annote = {Keywords: Covering, Extraction theorem, Double-disks, Submodularity, Local search} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n^{5/2} log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk^{3/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n/(k epsilon^5))*log^{5/2}(n/k)log((1/epsilon) log(n/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)k points of P, where k is the optimum value.
We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.

Haim Kaplan, Sasanka Roy, and Micha Sharir. Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 52:1-52:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kaplan_et_al:LIPIcs.ESA.2017.52, author = {Kaplan, Haim and Roy, Sasanka and Sharir, Micha}, title = {{Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {52:1--52:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.52}, URN = {urn:nbn:de:0030-drops-78608}, doi = {10.4230/LIPIcs.ESA.2017.52}, annote = {Keywords: Computational geometry, geometric optimization, rectangles, perimeter, area} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Asano et al. [JoCG 2011] proposed an open problem of computing the minimum enclosing circle of a set of n points in R^2 given in a read-only array in sub-quadratic time. We show that Megiddo's prune and search algorithm for computing the minimum radius circle enclosing the given points can be tailored to work in a read-only environment in O(n^{1+epsilon}) time using O(log n) extra space, where epsilon is a positive constant less than 1. As a warm-up, we first solve the same problem in an in-place setup in linear time with O(1) extra space.

Minati De, Subhas C. Nandy, and Sasanka Roy. Minimum Enclosing Circle with Few Extra Variables. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 510-521, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{de_et_al:LIPIcs.FSTTCS.2012.510, author = {De, Minati and Nandy, Subhas C. and Roy, Sasanka}, title = {{Minimum Enclosing Circle with Few Extra Variables}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {510--521}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.510}, URN = {urn:nbn:de:0030-drops-38855}, doi = {10.4230/LIPIcs.FSTTCS.2012.510}, annote = {Keywords: Minimum enclosing circle, space-efficient algorithm, prune-and-search} }

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