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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present a data structure that maintains a constant-factor approximate maximum independent set for broad classes of fat objects in d dimensions, where d is assumed to be a constant, in sublinear worst-case update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic (4+ε)-approximation for squares, with O(log⁴ n) worst-case update time.
Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with amortized update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem.
Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.

Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-Case Efficient Dynamic Geometric Independent Set. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2021.25, author = {Cardinal, Jean and Iacono, John and Koumoutsos, Grigorios}, title = {{Worst-Case Efficient Dynamic Geometric Independent Set}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {25:1--25:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.25}, URN = {urn:nbn:de:0030-drops-146061}, doi = {10.4230/LIPIcs.ESA.2021.25}, annote = {Keywords: Maximum independent set, deamortization, approximation} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We consider the online Min-Sum Set Cover (MSSC), a natural and intriguing generalization of the classical list update problem. In Online MSSC, the algorithm maintains a permutation on n elements based on subsets S₁, S₂, … arriving online. The algorithm serves each set S_t upon arrival, using its current permutation π_t, incurring an access cost equal to the position of the first element of S_t in π_t. Then, the algorithm may update its permutation to π_{t+1}, incurring a moving cost equal to the Kendall tau distance of π_t to π_{t+1}. The objective is to minimize the total access and moving cost for serving the entire sequence. We consider the r-uniform version, where each S_t has cardinality r. List update is the special case where r = 1.
We obtain tight bounds on the competitive ratio of deterministic online algorithms for MSSC against a static adversary, that serves the entire sequence by a single permutation. First, we show a lower bound of (r+1)(1-r/(n+1)) on the competitive ratio. Then, we consider several natural generalizations of successful list update algorithms and show that they fail to achieve any interesting competitive guarantee. On the positive side, we obtain a O(r)-competitive deterministic algorithm using ideas from online learning and the multiplicative weight updates (MWU) algorithm.
Furthermore, we consider efficient algorithms. We propose a memoryless online algorithm, called Move-All-Equally, which is inspired by the Double Coverage algorithm for the k-server problem. We show that its competitive ratio is Ω(r²) and 2^{O(√{log n ⋅ log r})}, and conjecture that it is f(r)-competitive. We also compare Move-All-Equally against the dynamic optimal solution and obtain (almost) tight bounds by showing that it is Ω(r √n) and O(r^{3/2} √n)-competitive.

Dimitris Fotakis, Loukas Kavouras, Grigorios Koumoutsos, Stratis Skoulakis, and Manolis Vardas. The Online Min-Sum Set Cover Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fotakis_et_al:LIPIcs.ICALP.2020.51, author = {Fotakis, Dimitris and Kavouras, Loukas and Koumoutsos, Grigorios and Skoulakis, Stratis and Vardas, Manolis}, title = {{The Online Min-Sum Set Cover Problem}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {51:1--51:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.51}, URN = {urn:nbn:de:0030-drops-124582}, doi = {10.4230/LIPIcs.ICALP.2020.51}, annote = {Keywords: Online Algorithms, Competitive Analysis, Min-Sum Set Cover} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

We study dynamic planar point location in the External Memory Model or Disk Access Model (DAM). Previous work in this model achieves polylog query and polylog amortized update time. We present a data structure with O(log_B^2 N) query time and O(1/B^(1-epsilon) log_B N) amortized update time, where N is the number of segments, B the block size and epsilon is a small positive constant, under the assumption that all faces have constant size. This is a B^(1-epsilon) factor faster for updates than the fastest previous structure, and brings the cost of insertion and deletion down to subconstant amortized time for reasonable choices of N and B. Our structure solves the problem of vertical ray-shooting queries among a dynamic set of interior-disjoint line segments; this is well-known to solve dynamic planar point location for a connected subdivision of the plane with faces of constant size.

John Iacono, Ben Karsin, and Grigorios Koumoutsos. External Memory Planar Point Location with Fast Updates. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{iacono_et_al:LIPIcs.ISAAC.2019.58, author = {Iacono, John and Karsin, Ben and Koumoutsos, Grigorios}, title = {{External Memory Planar Point Location with Fast Updates}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {58:1--58:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.58}, URN = {urn:nbn:de:0030-drops-115548}, doi = {10.4230/LIPIcs.ISAAC.2019.58}, annote = {Keywords: point location, data structures, dynamic algorithms, computational geometry} }

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