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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

The Euclidean Steiner Minimal Tree problem takes as input a set P of points in the Euclidean plane and finds the minimum length network interconnecting all the points of P. In this paper, in continuation to the works of [Du et al., 1987] and [Weng and Booth, 1995], we study Euclidean Steiner Minimal Tree when P is formed by the vertices of a pair of regular, concentric and parallel n-gons.
We restrict our attention to the cases where the two polygons are not very close to each other. In such cases, we show that Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an explicit structure of a Euclidean Steiner minimal tree for P.
We also consider point sets P of size n where the number of input points not on the convex hull of P is f(n) ≤ n. We give an exact algorithm with running time 2^𝒪(f(n) log n) for such input point sets P. Note that when f(n) = 𝒪(n/(log n)), our algorithm runs in single-exponential time, and when f(n) = o(n) the running time is 2^o(n log n) which is better than the known algorithm in [Hwang et al., 1992].
We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P = NP [Garey et al., 1977]. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets [Scott Provan, 1988]. In this paper, we show that if the number of input points in P not belonging to the convex hull of P is 𝒪(log n), then an FPTAS exists for Euclidean Steiner Minimal Tree. In contrast, we show that for any ε ∈ (0,1], when there are Ω(n^ε) points not belonging to the convex hull of the input set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P = NP.

Anubhav Dhar, Soumita Hait, and Sudeshna Kolay. Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dhar_et_al:LIPIcs.ISAAC.2023.25, author = {Dhar, Anubhav and Hait, Soumita and Kolay, Sudeshna}, title = {{Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {25:1--25:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.25}, URN = {urn:nbn:de:0030-drops-193273}, doi = {10.4230/LIPIcs.ISAAC.2023.25}, annote = {Keywords: Steiner minimal tree, Euclidean Geometry, Almost Convex point sets, FPTAS, strong NP-completeness} }

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

The Terrain Guarding problem is a well-known variant of the famous Art Gallery problem. Only second to Art Gallery, it is the most well-studied visibility problem in Discrete and Computational Geometry, which has also attracted attention from the viewpoint of Parameterized complexity. In this paper, we focus on the parameterized complexity of Terrain Guarding (both discrete and continuous) with respect to two natural parameters. First we show that, when parameterized by the number r of reflex vertices in the input terrain, the problem has a polynomial kernel. We also show that, when parameterized by the number c of minima in the terrain, Discrete Orthogonal Terrain Guarding has an XP algorithm.

Akanksha Agrawal, Sudeshna Kolay, and Meirav Zehavi. Parameter Analysis for Guarding Terrains. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2020.4, author = {Agrawal, Akanksha and Kolay, Sudeshna and Zehavi, Meirav}, title = {{Parameter Analysis for Guarding Terrains}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {4:1--4:18}, 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.2020.4}, URN = {urn:nbn:de:0030-drops-122514}, doi = {10.4230/LIPIcs.SWAT.2020.4}, annote = {Keywords: Terrain Guarding, Reflex Vertices, Terrain Minima, FPT Algorithm, XP Algorithm, Kernelization} }

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

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset R⊆ V(G) of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from V⧵ R. The vertices of R are referred to as terminals and the vertices of V(G)⧵ R as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in n^{O(√{t+k})} time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time 2^{O(k)}n^{O(1)}. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs [Fomin et al., 2019]. We mention that the algorithmic results can be made to work for Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree on disk graphs parameterized by k is W[1]-hard.

Sujoy Bhore, Paz Carmi, Sudeshna Kolay, and Meirav Zehavi. Parameterized Study of Steiner Tree on Unit Disk Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhore_et_al:LIPIcs.SWAT.2020.13, author = {Bhore, Sujoy and Carmi, Paz and Kolay, Sudeshna and Zehavi, Meirav}, title = {{Parameterized Study of Steiner Tree on Unit Disk Graphs}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {13:1--13:18}, 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.2020.13}, URN = {urn:nbn:de:0030-drops-122607}, doi = {10.4230/LIPIcs.SWAT.2020.13}, annote = {Keywords: Unit Disk Graphs, FPT, Subexponential exact algorithms, NP-Hardness, W-Hardness} }

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

Given a symmetric l x l matrix M=(m_{i,j}) with entries in {0,1,*}, a graph G and a function L : V(G) - > 2^{[l]} (where [l] = {1,2,...,l}), a list M-partition of G with respect to L is a partition of V(G) into l parts, say, V_1, V_2, ..., V_l such that for each i,j in {1,2,...,l}, (i) if m_{i,j}=0 then for any u in V_i and v in V_j, uv not in E(G), (ii) if m_{i,j}=1 then for any (distinct) u in V_i and v in V_j, uv in E(G), (iii) for each v in V(G), if v in V_i then i in L(v). We consider the Deletion to List M-Partition problem that takes as input a graph G, a list function L:V(G) - > 2^[l] and a positive integer k. The aim is to determine whether there is a k-sized set S subseteq V(G) such that G-S has a list M-partition. Many important problems like Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, Multiway Cut and Deletion to List Homomorphism are special cases of the Deletion to List M-Partition problem. In this paper, we provide a classification of the parameterized complexity of Deletion to List M-Partition, parameterized by k, (a) when M is of order at most 3, and (b) when M is of order 4 with all diagonal entries belonging to {0,1}.

Akanksha Agrawal, Sudeshna Kolay, Jayakrishnan Madathil, and Saket Saurabh. Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.ISAAC.2019.41, author = {Agrawal, Akanksha and Kolay, Sudeshna and Madathil, Jayakrishnan and Saurabh, Saket}, title = {{Parameterized Complexity Classification of Deletion to List Matrix-Partition for Low-Order Matrices}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {41:1--41:14}, 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.2019.41}, URN = {urn:nbn:de:0030-drops-115372}, doi = {10.4230/LIPIcs.ISAAC.2019.41}, annote = {Keywords: list matrix partitions, parameterized classification, Almost 2-SAT, important separators, iterative compression} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

In this paper, we study the query complexity of parameterized decision and optimization versions of Hitting-Set. We also investigate the query complexity of Packing. In doing so, we use generalizations to hypergraphs of an earlier query model, known as BIS introduced by Beame et al. in ITCS'18. The query models considered are the GPIS and GPISE oracles. The GPIS and GPISE oracles are used for the decision and optimization versions of the problems, respectively. We use color coding and queries to the oracles to generate subsamples from the hypergraph, that retain some structural properties of the original hypergraph. We use the stability of the sunflowers in a non-trivial way to do so.

Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh. Parameterized Query Complexity of Hitting Set Using Stability of Sunflowers. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bishnu_et_al:LIPIcs.ISAAC.2018.25, author = {Bishnu, Arijit and Ghosh, Arijit and Kolay, Sudeshna and Mishra, Gopinath and Saurabh, Saket}, title = {{Parameterized Query Complexity of Hitting Set Using Stability of Sunflowers}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {25:1--25:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.25}, URN = {urn:nbn:de:0030-drops-99735}, doi = {10.4230/LIPIcs.ISAAC.2018.25}, annote = {Keywords: Query complexity, Hitting set, Parameterized complexity} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

The Metric Embedding problem takes as input two metric spaces (X,D_X) and (Y,D_Y), and a positive integer d. The objective is to determine whether there is an embedding F:X - > Y such that the distortion d_{F} <= d. Such an embedding is called a distortion d embedding. In parameterized complexity, the Metric Embedding problem is known to be W-hard and therefore, not expected to have an FPT algorithm. In this paper, we consider the Gen-Graph Metric Embedding problem, where the two metric spaces are graph metrics. We explore the extent of tractability of the problem in the parameterized complexity setting. We determine whether an unweighted graph metric (G,D_G) can be embedded, or bijectively embedded, into another unweighted graph metric (H,D_H), where the graph H has low structural complexity. For example, H is a cycle, or H has bounded treewidth or bounded connected treewidth. The parameters for the algorithms are chosen from the upper bound d on distortion, bound Delta on the maximum degree of H, treewidth alpha of H, and the connected treewidth alpha_{c} of H.
Our general approach to these problems can be summarized as trying to understand the behavior of the shortest paths in G under a low distortion embedding into H, and the structural relation the mapping of these paths has to shortest paths in H.

Arijit Ghosh, Sudeshna Kolay, and Gopinath Mishra. FPT Algorithms for Embedding into Low Complexity Graphic Metrics. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ghosh_et_al:LIPIcs.ESA.2018.35, author = {Ghosh, Arijit and Kolay, Sudeshna and Mishra, Gopinath}, title = {{FPT Algorithms for Embedding into Low Complexity Graphic Metrics}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {35:1--35:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.35}, URN = {urn:nbn:de:0030-drops-94988}, doi = {10.4230/LIPIcs.ESA.2018.35}, annote = {Keywords: Metric spaces, metric embedding, FPT, dynamic programming} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Given a graph G and a pair (\mathcal{F}_1,\mathcal{F}_2) of graph families, the function {\sf GDISJ}_{G,{\cal F}_1,{\cal F}_2} takes as input, two induced subgraphs G_1 and G_2 of G, such that G_1 \in \mathcal{F}_1 and G_2 \in \mathcal{F}_2 and returns 1 if V(G_1)\cap V(G_2)=\emptyset and 0 otherwise. We study the communication complexity of this problem in the two-party model. In particular, we look at pairs of hereditary graph families. We show that the communication complexity of this function, when the two graph families are hereditary, is sublinear if and only if there are finitely many graphs in the intersection of these two families. Then, using concepts from parameterized complexity, we
obtain nuanced upper bounds on the communication complexity of GDISJ_G,\cal F_1,\cal F_2. A concept related to communication protocols is that of a (\mathcal{F}_1,\mathcal{F}_2)-separating family of a graph G. A collection \mathcal{F} of subsets of V(G) is
called a (\mathcal{F}_1,\mathcal{F}_2)-separating family} for G, if for any two vertex disjoint induced subgraphs G_1\in \mathcal{F}_1,G_2\in \mathcal{F}_2, there is a set F \in \mathcal{F} with V(G_1) \subseteq F and V(G_2) \cap F = \emptyset.
Given a graph G on n vertices, for any pair (\mathcal{F}_1,\mathcal{F}_2) of hereditary graph families with sublinear communication complexity for GDISJ_G,\cal F_1,\cal F_2, we give an enumeration algorithm that finds a subexponential sized (\mathcal{F}_1,\mathcal{F}_2)-separating
family. In fact, we give an enumeration algorithm that finds a 2^{o(k)}n^{\Oh(1)} sized (\mathcal{F}_1,\mathcal{F}_2)-separating family; where k denotes the size of a minimum sized set S of vertices such that V(G)\setminus S has a bipartition (V_1,V_2) with G[V_1] \in {\cal F}_1 and G[V_2]\in {\cal F}_2. We exhibit a wide range of applications for these separating families, to obtain combinatorial bounds, enumeration algorithms as well as exact and FPT algorithms for several problems.

Sudeshna Kolay, Fahad Panolan, and Saket Saurabh. Communication Complexity of Pairs of Graph Families with Applications. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kolay_et_al:LIPIcs.MFCS.2017.13, author = {Kolay, Sudeshna and Panolan, Fahad and Saurabh, Saket}, title = {{Communication Complexity of Pairs of Graph Families with Applications}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.13}, URN = {urn:nbn:de:0030-drops-80849}, doi = {10.4230/LIPIcs.MFCS.2017.13}, annote = {Keywords: Communication Complexity, Separating Family, FPT algorithms} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in \mathbb{R}^d and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in
general position with respect to a subsystem of hyperplanes in \mathbb{R}^d if no d+1 points lie on the same
hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations.
When parameterized by k then we exhibit a polynomial kernelization for the problem. When parameterized by h=n-k,
or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic
assumptions.
We can also exhibit similar kernelization results for d-Polynomial Subset General Position, where a vector space of polynomials
of degree at most d are specified as the underlying subsystem such that the size of the basis for this vector space is b. The objective is to find a set of at least k input points, or in the dual delete at most h = n-k points, such that no b+1 points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position. We also study general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem.

Jean-Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, and Sudeshna Kolay. Kernelization of the Subset General Position Problem in Geometry. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{boissonnat_et_al:LIPIcs.MFCS.2017.25, author = {Boissonnat, Jean-Daniel and Dutta, Kunal and Ghosh, Arijit and Kolay, Sudeshna}, title = {{Kernelization of the Subset General Position Problem in Geometry}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {25:1--25:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.25}, URN = {urn:nbn:de:0030-drops-80863}, doi = {10.4230/LIPIcs.MFCS.2017.25}, annote = {Keywords: Incidence Geometry, Kernel Lower bounds, Hyperplanes, Bounded degree polynomials} }

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

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.

Pradeesha Ashok, Fedor V. Fomin, Sudeshna Kolay, Saket Saurabh, and Meirav Zehavi. Exact Algorithms for Terrain Guarding. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ashok_et_al:LIPIcs.SoCG.2017.11, author = {Ashok, Pradeesha and Fomin, Fedor V. and Kolay, Sudeshna and Saurabh, Saket and Zehavi, Meirav}, title = {{Exact Algorithms for Terrain Guarding}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {11:1--11:15}, 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.2017.11}, URN = {urn:nbn:de:0030-drops-71975}, doi = {10.4230/LIPIcs.SoCG.2017.11}, annote = {Keywords: Terrain Guarding, Art Gallery, Exponential-Time Algorithms} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

For fixed integers r,l >= 0, a graph G is called an (r,l)-graph if the vertex set V(G) can be partitioned into r independent sets and l cliques. Such a graph is also said to have cochromatic number r+l. The class of (r,l) graphs generalizes r-colourable graphs (when l=0) and hence not surprisingly, determining whether a given graph is an (r,l)-graph is NP-hard even when r >= 3 or l >= 3 in general graphs.
When r and ell are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the Chromatic Number problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by r and l. I.e. there is an f(r+l) n^O(1) algorithm on perfect graphs on n vertices where f is a function of r and l. Observe that such an algorithm is unlikely on general graphs as the problem is NP-hard even for constant r and l.
In this paper, we consider the parameterized complexity of the following problem, which we call Vertex Partization. Given a perfect graph G and positive integers r,l,k decide whether there exists a set S subset or equal to V(G) of size at most k such that the deletion of S from G results in an (r,l)-graph. This problem generalizes well studied problems such as Vertex Cover (when r=1 and l=0), Odd Cycle Transversal (when r=2, l=0) and Split Vertex Deletion (when r=1=l).
1. Vertex Partization on perfect graphs is FPT when parameterized by k+r+l.
2. The problem, when parameterized by k+r+l, does not admit any polynomial sized kernel, under standard complexity theoretic assumptions. In other words, in polynomial time, the input graph cannot be compressed to an equivalent instance of size polynomial in k+r+l. In fact, our result holds even when k=0.
3. When r,ell are universal constants, then Vertex Partization on perfect graphs, parameterized by k, has a polynomial sized kernel.

Sudeshna Kolay, Fahad Panolan, Venkatesh Raman, and Saket Saurabh. Parameterized Algorithms on Perfect Graphs for Deletion to (r,l)-Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 75:1-75:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kolay_et_al:LIPIcs.MFCS.2016.75, author = {Kolay, Sudeshna and Panolan, Fahad and Raman, Venkatesh and Saurabh, Saket}, title = {{Parameterized Algorithms on Perfect Graphs for Deletion to (r,l)-Graphs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {75:1--75:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.75}, URN = {urn:nbn:de:0030-drops-64846}, doi = {10.4230/LIPIcs.MFCS.2016.75}, annote = {Keywords: graph deletion, FPT algorithms, polynomial kernels} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

Fedor Fomin, Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fomin_et_al:LIPIcs.SoCG.2016.39, author = {Fomin, Fedor and Kolay, Sudeshna and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket}, title = {{Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {39:1--39:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.39}, URN = {urn:nbn:de:0030-drops-59310}, doi = {10.4230/LIPIcs.SoCG.2016.39}, annote = {Keywords: Rectilinear graphs, Steiner arborescence, parameterized algorithms} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results:
- We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems.
- Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open.
For our approximation algorithms and thus for Turing kernels we use
an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

Sudeshna Kolay and Fahad Panolan. Parameterized Algorithms for Deletion to (r,ell)-Graphs. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 420-433, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kolay_et_al:LIPIcs.FSTTCS.2015.420, author = {Kolay, Sudeshna and Panolan, Fahad}, title = {{Parameterized Algorithms for Deletion to (r,ell)-Graphs}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {420--433}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.420}, URN = {urn:nbn:de:0030-drops-56456}, doi = {10.4230/LIPIcs.FSTTCS.2015.420}, annote = {Keywords: FPT, Turing kernels, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G\S belongs to F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C_{odd}, families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C_{odd} are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12^{k}*n^{O(1)} for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.

Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Quick but Odd Growth of Cacti. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 258-269, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kolay_et_al:LIPIcs.IPEC.2015.258, author = {Kolay, Sudeshna and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket}, title = {{Quick but Odd Growth of Cacti}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {258--269}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.258}, URN = {urn:nbn:de:0030-drops-55883}, doi = {10.4230/LIPIcs.IPEC.2015.258}, annote = {Keywords: Even Cycle Transversal, Diamond Hitting Set, Randomized Algorithms} }

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