Document

**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Let P be a set of at most n points and let R be a set of at most n geometric ranges, such as disks and rectangles, where each p ∈ P has an associated supply s_{p} > 0, and each r ∈ R has an associated demand d_r > 0. A (many-to-many) matching is a set 𝒜 of ordered triples (p,r,a_{pr}) ∈ P × R × ℝ_{> 0} such that p ∈ r and the a_{pr}’s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing ∑_{(p,r,a_{pr}) ∈ 𝒜} a_{pr}.
Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of n red points P and a set of n blue points Q that minimizes the length of the longest edge. For the L_∞-metric, we can do this in time O(n^{1+ε}) in any fixed dimension, for the L₂-metric in the plane in time O(n^{4/3 + ε}), for any ε > 0.

Sergio Cabello, Siu-Wing Cheng, Otfried Cheong, and Christian Knauer. Geometric Matching and Bottleneck Problems. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2024.31, author = {Cabello, Sergio and Cheng, Siu-Wing and Cheong, Otfried and Knauer, Christian}, title = {{Geometric Matching and Bottleneck Problems}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {31:1--31:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.31}, URN = {urn:nbn:de:0030-drops-199768}, doi = {10.4230/LIPIcs.SoCG.2024.31}, annote = {Keywords: Many-to-many matching, bipartite, planar, geometric, approximation} }

Document

**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number.
We show that the page number of an ordered graph with n vertices and m edges can be computed in 2^m ⋅ n^𝒪(1) time. An 𝒪(log n)-approximation of this number can be computed efficiently. We can decide in 2^𝒪(d √k log (d+k)) ⋅ n^𝒪(1) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h = 1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h > 1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in 2ⁿ ⋅ n^𝒪(1) time either the number of crossings or, if we disallow crossings, the number of tracks.

Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating Crossings in Ordered Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2024.1, author = {Agrawal, Akanksha and Cabello, Sergio and Kaufmann, Michael and Saurabh, Saket and Sharma, Roohani and Uno, Yushi and Wolff, Alexander}, title = {{Eliminating Crossings in Ordered Graphs}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.1}, URN = {urn:nbn:de:0030-drops-200417}, doi = {10.4230/LIPIcs.SWAT.2024.1}, annote = {Keywords: Ordered graphs, book embedding, edge deletion, d-planar, hitting set} }

Document

**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We study the problem of k-means clustering in the space of straight-line segments in ℝ² under the Hausdorff distance. For this problem, we give a (1+ε)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n + ε^{-O(k)} + ε^{-O(k)} ⋅ log^O(k) (ε^{-1})). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron [Antoine Vigneron, 2014] to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg [Dan Feldman and Michael Langberg, 2011; Feldman et al., 2020]. Our results can be extended to polylines of constant complexity with a running time of O(n + ε^{-O(k)}).

Sergio Cabello and Panos Giannopoulos. On k-Means for Segments and Polylines. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 28:1-28:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cabello_et_al:LIPIcs.ESA.2023.28, author = {Cabello, Sergio and Giannopoulos, Panos}, title = {{On k-Means for Segments and Polylines}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {28:1--28:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.28}, URN = {urn:nbn:de:0030-drops-186812}, doi = {10.4230/LIPIcs.ESA.2023.28}, annote = {Keywords: k-means clustering, segments, polylines, Hausdorff distance, Fr\'{e}chet mean} }

Document

**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

In the longest plane spanning tree problem, we are given a finite planar point set 𝒫, and our task is to find a plane (i.e., noncrossing) spanning tree T_OPT for 𝒫 with maximum total Euclidean edge length |T_OPT|. Despite more than two decades of research, it remains open if this problem is NP-hard. Thus, previous efforts have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates |T_OPT|. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms.
We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree T_ALG with diameter at most four and |T_ALG| ≥ 0.546 ⋅ |T_OPT|. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter d ≥ 3, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most d (compared to a longest plane tree without constraints).

Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long Plane Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2022.23, author = {Cabello, Sergio and Hoffmann, Michael and Klost, Katharina and Mulzer, Wolfgang and Tkadlec, Josef}, title = {{Long Plane Trees}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.23}, URN = {urn:nbn:de:0030-drops-160311}, doi = {10.4230/LIPIcs.SoCG.2022.23}, annote = {Keywords: geometric network design, spanning trees, plane straight-line graphs, approximation algorithms} }

Document

Invited Talk

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

In this paper we shall encounter three open problems in Computational Geometry that are, in my opinion, interesting for a general audience interested in algorithms.

Sergio Cabello. Some Open Problems in Computational Geometry (Invited Talk). In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 2:1-2:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cabello:LIPIcs.MFCS.2020.2, author = {Cabello, Sergio}, title = {{Some Open Problems in Computational Geometry}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {2:1--2:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.2}, URN = {urn:nbn:de:0030-drops-126734}, doi = {10.4230/LIPIcs.MFCS.2020.2}, annote = {Keywords: barrier resilience, maximum matching, geometric graphs, fixed-parameter tractability, stochastic computational geometry} }

Document

Complete Volume

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

LIPIcs, Volume 164, SoCG 2020, Complete Volume

36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1-1222, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{cabello_et_al:LIPIcs.SoCG.2020, title = {{LIPIcs, Volume 164, SoCG 2020, Complete Volume}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {1--1222}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020}, URN = {urn:nbn:de:0030-drops-121576}, doi = {10.4230/LIPIcs.SoCG.2020}, annote = {Keywords: LIPIcs, Volume 164, SoCG 2020, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Front Matter, Table of Contents, Preface, Conference Organization

36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2020.0, author = {Cabello, Sergio and Chen, Danny Z.}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {0:i--0:xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.0}, URN = {urn:nbn:de:0030-drops-121587}, doi = {10.4230/LIPIcs.SoCG.2020.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O(ρ^{3ω/2}n^{ω/2}) time with high probability, where ρ is the density of the geometric objects and ω>2 is a constant such that n × n matrices can be multiplied in O(n^ω) time.
The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators.
We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O(n^{ω/2}) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [1, Ψ] can be found in O(Ψ⁶log^11 n + Ψ^{12 ω} n^{ω/2}) time with high probability.

Édouard Bonnet, Sergio Cabello, and Wolfgang Mulzer. Maximum Matchings in Geometric Intersection Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonnet_et_al:LIPIcs.STACS.2020.31, author = {Bonnet, \'{E}douard and Cabello, Sergio and Mulzer, Wolfgang}, title = {{Maximum Matchings in Geometric Intersection Graphs}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {31:1--31:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.31}, URN = {urn:nbn:de:0030-drops-118926}, doi = {10.4230/LIPIcs.STACS.2020.31}, annote = {Keywords: computational geometry, geometric intersection graph, maximum matching, disk graph, unit-disk graph} }

Document

**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box.
For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods.
We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points.

Sergio Cabello and Timothy M. Chan. Computing Shapley Values in the Plane. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2019.20, author = {Cabello, Sergio and Chan, Timothy M.}, title = {{Computing Shapley Values in the Plane}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.20}, URN = {urn:nbn:de:0030-drops-104244}, doi = {10.4230/LIPIcs.SoCG.2019.20}, annote = {Keywords: Shapley values, stochastic computational geometry, convex hull, minimum enclosing disk, bounding box, arrangements, convolutions, airport problem} }

Document

**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We prove a generalization of Pál's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q. We also prove a lower bound of Omega(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, and Sang Duk Yoon. The Reverse Kakeya Problem. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bae_et_al:LIPIcs.SoCG.2018.6, author = {Bae, Sang Won and Cabello, Sergio and Cheong, Otfried and Choi, Yoonsung and Stehn, Fabian and Yoon, Sang Duk}, title = {{The Reverse Kakeya Problem}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {6:1--6:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.6}, URN = {urn:nbn:de:0030-drops-87199}, doi = {10.4230/LIPIcs.SoCG.2018.6}, annote = {Keywords: Kakeya problem, convex, isodynamic point, turning} }

Document

**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that:
* The problem is NP-hard already in 3 dimensions.
* In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm.
* For any constant dimension d, we give an efficient polynomial-time approximation scheme.

Karl Bringmann, Sergio Cabello, and Michael T. M. Emmerich. Maximum Volume Subset Selection for Anchored Boxes. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2017.22, author = {Bringmann, Karl and Cabello, Sergio and Emmerich, Michael T. M.}, title = {{Maximum Volume Subset Selection for Anchored Boxes}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {22:1--22:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.22}, URN = {urn:nbn:de:0030-drops-72011}, doi = {10.4230/LIPIcs.SoCG.2017.22}, annote = {Keywords: geometric optimization, subset selection, hypervolume indicator, Klee’s 23 measure problem, boxes, NP-hardness, PTAS} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^(o(sqrt(k)))) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).

Benjamin Burton, Sergio Cabello, Stefan Kratsch, and William Pettersson. The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{burton_et_al:LIPIcs.STACS.2017.18, author = {Burton, Benjamin and Cabello, Sergio and Kratsch, Stefan and Pettersson, William}, title = {{The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.18}, URN = {urn:nbn:de:0030-drops-70156}, doi = {10.4230/LIPIcs.STACS.2017.18}, annote = {Keywords: computational topology, parameterized complexity, simplicial complex} }

Document

**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results:
- In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1.
- In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1.
- In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction.
The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.

Drago Bokal, Sergio Cabello, and David Eppstein. Finding All Maximal Subsequences with Hereditary Properties. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 240-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bokal_et_al:LIPIcs.SOCG.2015.240, author = {Bokal, Drago and Cabello, Sergio and Eppstein, David}, title = {{Finding All Maximal Subsequences with Hereditary Properties}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {240--254}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.240}, URN = {urn:nbn:de:0030-drops-51132}, doi = {10.4230/LIPIcs.SOCG.2015.240}, annote = {Keywords: convex hull, diameter, monotone path, sequence of points, trajectory} }

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