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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions.
First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ℝ^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an Ω((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete.
Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ℝ^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ≥ d+1 finite point sets in ℝ^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d.

Helena Bergold, Daniel Bertschinger, Nicolas Grelier, Wolfgang Mulzer, and Patrick Schnider. Well-Separation and Hyperplane Transversals in High Dimensions. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bergold_et_al:LIPIcs.SWAT.2022.16, author = {Bergold, Helena and Bertschinger, Daniel and Grelier, Nicolas and Mulzer, Wolfgang and Schnider, Patrick}, title = {{Well-Separation and Hyperplane Transversals in High Dimensions}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.16}, URN = {urn:nbn:de:0030-drops-161766}, doi = {10.4230/LIPIcs.SWAT.2022.16}, annote = {Keywords: hyperplane transversal, high-dimension, hardness} }

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**Published in:** LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Let 𝒟 be a set of straight-line segments in the plane, potentially crossing, and let c be a positive integer. We denote by P the union of the endpoints of the straight-line segments of 𝒟 and of the intersection points between pairs of segments. We say that 𝒟 has a nearest-neighbor decomposition into c parts if we can partition P into c point sets P₁, … , P_c such that 𝒟 is the union of the nearest neighbor graphs on P₁, … , P_c. We show that it is NP-complete to decide whether 𝒟 can be drawn as the union of c ≥ 3 nearest-neighbor graphs, even when no two segments cross. We show that for c = 2, it is NP-complete in the general setting and polynomial-time solvable when no two segments cross. We show the existence of an O(log n)-approximation algorithm running in subexponential time for partitioning 𝒟 into a minimum number of nearest-neighbor graphs.
As a main tool in our analysis, we establish the notion of the conflict graph for a drawing 𝒟. The vertices of the conflict graph are the connected components of 𝒟, with the assumption that each connected component is the nearest neighbor graph of its vertices, and there is an edge between two components U and V if and only if the nearest neighbor graph of U ∪ V contains an edge between a vertex in U and a vertex in V. We show that string graphs are conflict graphs of certain planar drawings. For planar graphs and complete k-partite graphs, we give additional, more efficient constructions. We furthermore show that there are subdivisions of non-planar graphs that are not conflict graphs. Lastly, we show a separator lemma for conflict graphs.

Jonas Cleve, Nicolas Grelier, Kristin Knorr, Maarten Löffler, Wolfgang Mulzer, and Daniel Perz. Nearest-Neighbor Decompositions of Drawings. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 21:1-21:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cleve_et_al:LIPIcs.SWAT.2022.21, author = {Cleve, Jonas and Grelier, Nicolas and Knorr, Kristin and L\"{o}ffler, Maarten and Mulzer, Wolfgang and Perz, Daniel}, title = {{Nearest-Neighbor Decompositions of Drawings}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.21}, URN = {urn:nbn:de:0030-drops-161812}, doi = {10.4230/LIPIcs.SWAT.2022.21}, annote = {Keywords: nearest-neighbors, decompositions, drawing} }

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

We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an 𝒪(log OPT)-approximation running in 𝒪(n⁸)-time, where OPT denotes the minimum number of convex faces needed, to an 𝒪(√nlog n)-approximation algorithm running in 𝒪(n²)-time. We say that a point set is k-directed if the (straight) lines containing at least three points have up to k directions. We present an 𝒪(k)-approximation algorithm running in n^{𝒪(k)}-time. Those hardness and approximation results also holds for the Minimum Convex Tiling problem, defined similarly but allowing the use of Steiner points. The approximation results are obtained by relating the problem to the Covering Points with Non-Crossing Segments problem. We show that this problem is NP-hard, and present an FPT algorithm. This allows us to obtain a constant-approximation FPT algorithm for the Minimum Convex Partition Problem where the parameter is the number of faces.

Nicolas Grelier. Hardness and Approximation of Minimum Convex Partition. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 45:1-45:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{grelier:LIPIcs.SoCG.2022.45, author = {Grelier, Nicolas}, title = {{Hardness and Approximation of Minimum Convex Partition}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {45:1--45:15}, 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.2022.45}, URN = {urn:nbn:de:0030-drops-160530}, doi = {10.4230/LIPIcs.SoCG.2022.45}, annote = {Keywords: degenerate point sets, point cover, non-crossing segments, approximation algorithm, complexity} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark '90, Raghavan and Spinrad '03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. '18, Bonamy et al. '18] to homothets of a fixed centrally symmetric convex set.
The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes I follow the same road. They show that, for every graph G of a large-enough class C, the complement of an even subdivision of G belongs to the intersection class I. Then they conclude by invoking the hardness of Maximum Independent Set on the class C, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. '18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.

Édouard Bonnet, Nicolas Grelier, and Tillmann Miltzow. Maximum Clique in Disk-Like Intersection Graphs. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 17:1-17:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonnet_et_al:LIPIcs.FSTTCS.2020.17, author = {Bonnet, \'{E}douard and Grelier, Nicolas and Miltzow, Tillmann}, title = {{Maximum Clique in Disk-Like Intersection Graphs}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.17}, URN = {urn:nbn:de:0030-drops-132587}, doi = {10.4230/LIPIcs.FSTTCS.2020.17}, annote = {Keywords: Disk Graphs, Intersection Graphs, Maximum Clique, Algorithms, NP-hardness, APX-hardness} }

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