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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

A disk graph is an intersection graph of disks in the Euclidean plane, where the disks correspond to the vertices of the graph and a pair of vertices are adjacent if and only if their corresponding disks intersect. The problem of determining the time complexity of computing a maximum clique in a disk graph is a long-standing open question that has been very well studied in the literature. The problem is known to be open even when the radii of all the disks are in the interval [1,(1+ε)], where ε > 0. If all the disks are unit disks then there exists an O(n³log n)-time algorithm to compute a maximum clique, which is the best-known running time for over a decade. Although the problem of computing a maximum clique in a disk graph remains open, it is known to be APX-hard for the intersection graphs of many other convex objects such as intersection graphs of ellipses, triangles, and a combination of unit disks and axis-parallel rectangles. Here we obtain the following results.
- We give an algorithm to compute a maximum clique in a unit disk graph in O(n^2.5 log n)-time, which improves the previously best known running time of O(n³log n) [Eppstein '09].
- We extend a widely used "co-2-subdivision approach" to prove that computing a maximum clique in a combination of unit disks and axis-parallel rectangles is NP-hard to approximate within 4448/4449 ≈ 0.9997. The use of a "co-2-subdivision approach" was previously thought to be unlikely in this setting [Bonnet et al. '20]. Our result improves the previously known inapproximability factor of 7633010347/7633010348 ≈ 0.9999.
- We show that the parameter minimum lens width of the disk arrangement may be used to make progress in the case when disk radii are in [1,(1+ε)]. For example, if the minimum lens width is at least 0.265 and ε ≤ 0.0001, which still allows for non-Helly triples in the arrangement, then one can find a maximum clique in polynomial time.

Jared Espenant, J. Mark Keil, and Debajyoti Mondal. Finding a Maximum Clique in a Disk Graph. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{espenant_et_al:LIPIcs.SoCG.2023.30, author = {Espenant, Jared and Keil, J. Mark and Mondal, Debajyoti}, title = {{Finding a Maximum Clique in a Disk Graph}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.30}, URN = {urn:nbn:de:0030-drops-178803}, doi = {10.4230/LIPIcs.SoCG.2023.30}, annote = {Keywords: Maximum clique, Disk graph, Time complexity, APX-hardness} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

A 2-interval is the union of two disjoint intervals on the real line. Two 2-intervals D₁ and D₂ are disjoint if their intersection is empty (i.e., no interval of D₁ intersects any interval of D₂). There can be three different relations between two disjoint 2-intervals; namely, preceding (<), nested (⊏) and crossing (≬). Two 2-intervals D₁ and D₂ are called R-comparable for some R∈{<,⊏,≬}, if either D₁RD₂ or D₂RD₁. A set 𝒟 of disjoint 2-intervals is ℛ-comparable, for some ℛ⊆{<,⊏,≬} and ℛ≠∅, if every pair of 2-intervals in ℛ are R-comparable for some R∈ℛ. Given a set of 2-intervals and some ℛ⊆{<,⊏,≬}, the objective of the {2-interval pattern problem} is to find a largest subset of 2-intervals that is ℛ-comparable.
The 2-interval pattern problem is known to be W[1]-hard when |ℛ|=3 and NP-hard when |ℛ|=2 (except for ℛ={<,⊏}, which is solvable in quadratic time). In this paper, we fully settle the parameterized complexity of the problem by showing that it is W[1]-hard for both ℛ={⊏,≬} and ℛ={<,≬} (when parameterized by the size of an optimal solution). This answers the open question posed by Vialette [Encyclopedia of Algorithms, 2008].

Prosenjit Bose, Saeed Mehrabi, and Debajyoti Mondal. Parameterized Complexity of Two-Interval Pattern Problem. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 16:1-16:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2020.16, author = {Bose, Prosenjit and Mehrabi, Saeed and Mondal, Debajyoti}, title = {{Parameterized Complexity of Two-Interval Pattern Problem}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {16:1--16:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.16}, URN = {urn:nbn:de:0030-drops-122630}, doi = {10.4230/LIPIcs.SWAT.2020.16}, annote = {Keywords: Interval graphs, Two-interval pattern problem, Comparability, Multicoloured clique problem, Parameterized complexity, W\lbrack1\rbrack-hardness} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Given a set of n points (sites) inside a rectangle R and n points (label locations or ports) on its boundary, a boundary labeling problem seeks ways of connecting every site to a distinct port while achieving different labeling aesthetics. We examine the scenario when the connecting lines (leaders) are drawn as axis-aligned polylines with few bends, every leader lies strictly inside R, no two leaders cross, and the sum of the lengths of all the leaders is minimized. In a k-sided boundary labeling problem, where 1 <= k <= 4, the label locations are located on the k consecutive sides of R.
In this paper we develop an O(n^3 log n)-time algorithm for 2-sided boundary labeling, where the leaders are restricted to have one bend. This improves the previously best known O(n^8 log n)-time algorithm of Kindermann et al. (Algorithmica, 76(1):225-258, 2016). We show the problem is polynomial-time solvable in more general settings such as when the ports are located on more than two sides of R, in the presence of obstacles, and even when the objective is to minimize the total number of bends. Our results improve the previous algorithms on boundary labeling with obstacles, as well as provide the first polynomial-time algorithms for minimizing the total leader length and number of bends for 3- and 4-sided boundary labeling. These results settle a number of open questions on the boundary labeling problems (Wolff, Handbook of Graph Drawing, Chapter 23, Table 23.1, 2014).

Prosenjit Bose, Paz Carmi, J. Mark Keil, Saeed Mehrabi, and Debajyoti Mondal. Boundary Labeling for Rectangular Diagrams. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bose_et_al:LIPIcs.SWAT.2018.12, author = {Bose, Prosenjit and Carmi, Paz and Keil, J. Mark and Mehrabi, Saeed and Mondal, Debajyoti}, title = {{Boundary Labeling for Rectangular Diagrams}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.12}, URN = {urn:nbn:de:0030-drops-88386}, doi = {10.4230/LIPIcs.SWAT.2018.12}, annote = {Keywords: Boundary labeling, Dynamic programming, Outerstring graphs} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The thickness of a graph G = (V, E) with n vertices is the minimum number of planar subgraphs of G whose union is G. A polyline drawing of G in R^2 is a drawing Gamma of G, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Gamma is the maximum number of bends per edge in Gamma, and the layer complexity of Gamma is the minimum integer r such that the set of polygonal chains in Gamma can be partitioned into r disjoint sets, where each set corresponds to a planar polyline drawing. Let G be a graph of thickness t. By Fáry’s theorem, if t = 1, then G can be drawn on a single layer with bend complexity 0. A few extensions to higher thickness are known, e.g., if t = 2 (resp., t > 2), then G can be drawn on t layers with bend complexity 2 (resp., 3n + O(1)).
In this paper we present an elegant extension of Fáry's theorem to draw graphs of thickness t > 2. We first prove that thickness-t graphs can be drawn on t layers with 2.25n + O(1) bends per edge. We then develop another technique to draw thickness-t graphs on t layers with reduced bend complexity for small values of t, e.g., for t in {3, 4}, the bend complexity decreases to O(sqrt(n)).
Previously, the bend complexity was not known to be sublinear for t > 2. Finally, we show that graphs with linear arboricity k can be drawn on k layers with bend complexity 3*(k-1)*n/(4k-2).

Stephane Durocher and Debajyoti Mondal. Relating Graph Thickness to Planar Layers and Bend Complexity. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{durocher_et_al:LIPIcs.ICALP.2016.10, author = {Durocher, Stephane and Mondal, Debajyoti}, title = {{Relating Graph Thickness to Planar Layers and Bend Complexity}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.10}, URN = {urn:nbn:de:0030-drops-62767}, doi = {10.4230/LIPIcs.ICALP.2016.10}, annote = {Keywords: Graph Drawing, Thickness, Geometric Thickness, Layers; Bends} }