32 Search Results for "Varadarajan, Kasturi"

Document
DOI: 10.4230/LIPIcs.STACS.2026.13
Line Cover and Related Problems

Authors: Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We study several extensions of the classic Line Cover problem of covering a set of n points in the plane with k lines. Line Cover is known to be NP-hard and our focus is on two natural generalizations: (1) Line Clustering, where the objective is to find k lines in the plane that minimize the sum of squares of distances of a given set of input points to the closest line, and (2) Hyperplane Cover, where the goal is to cover n points in ℝ^d by k hyperplanes. We also consider the more general Projective Clustering problem, which unifies both of these and has numerous applications in machine learning, data mining, and computational geometry. In this problem one seeks k affine subspaces of dimension r minimizing the sum of squares of distances of a given set of n points in ℝ^d to the closest point within one of the k affine subspaces. Our main contributions reveal interesting differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable parameterized by the number k of lines in the solution, we show that Line Clustering is W[1]-hard when parameterized by k and rule out algorithms of running time n^{o(k)} under the Exponential Time Hypothesis. Hyperplane Cover is known to be NP-hard even when d = 2 and by the work of Langerman and Morin [Discrete & Computational Geometry, 2005], it is FPT parameterized by k and d. We complement this result by establishing that Hyperplane Cover is W[2]-hard when parameterized by only k. We complement our hardness results by presenting an algorithm for Projective Clustering. We show that this problem is solvable in n^{𝒪(dk(r+1))} time. Not only does this yield an upper bound for Line Clustering that asymptotically matches our lower bound, but it also significantly extends the seminal work on k-Means Clustering (the special case r = 0) by Inaba, Katoh, and Imai [SoCG 1994].
Cite as

Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana. Line Cover and Related Problems. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bentert_et_al:LIPIcs.STACS.2026.13,
  author =	{Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A. and Saha, Souvik and Seetharaman, Sanjay and Upasana, Anannya},
  title =	{{Line Cover and Related Problems}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.13},
  URN =		{urn:nbn:de:0030-drops-255023},
  doi =		{10.4230/LIPIcs.STACS.2026.13},
  annote =	{Keywords: Point Line Cover, Projective Clustering, W-hardness, XP algorithm}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.8
Binary k-Center with Missing Entries: Structure Leads to Tractability

Authors: Tobias Friedrich, Kirill Simonov, and Farehe Soheil

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
k-Center clustering is a fundamental classification problem, where the task is to categorize the given collection of entities into k clusters and come up with a representative for each cluster, so that the maximum distance between an entity and its representative is minimized. In this work, we focus on the setting where the entities are represented by binary vectors with missing entries, which model incomplete categorical data. This version of the problem has wide applications, from predictive analytics to bioinformatics. Our main finding is that the problem, which is notoriously hard from the classical complexity viewpoint, becomes tractable as soon as the known entries are sparse and exhibit a certain structure. Formally, we show fixed-parameter tractable algorithms for the parameters vertex cover, fracture number, and treewidth of the row-column graph, which encodes the positions of the known entries of the matrix. Additionally, we tie the complexity of the 1-cluster variant of the problem, which is famous under the name Closest String, to the complexity of solving integer linear programs with few constraints. This implies, in particular, that improving upon the running times of our algorithms would lead to more efficient algorithms for integer linear programming in general.
Cite as

Tobias Friedrich, Kirill Simonov, and Farehe Soheil. Binary k-Center with Missing Entries: Structure Leads to Tractability. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{friedrich_et_al:LIPIcs.IPEC.2025.8,
  author =	{Friedrich, Tobias and Simonov, Kirill and Soheil, Farehe},
  title =	{{Binary k-Center with Missing Entries: Structure Leads to Tractability}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.8},
  URN =		{urn:nbn:de:0030-drops-251403},
  doi =		{10.4230/LIPIcs.IPEC.2025.8},
  annote =	{Keywords: Clustering, Missing Entries, k-Center, Parameterized Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.46
Max-Distance Sparsification for Diversification and Clustering

Authors: Soh Kumabe

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Let 𝒟 be a set family that is the solution domain of some combinatorial problem. The max-min diversification problem on 𝒟 is the problem to select k sets from 𝒟 such that the Hamming distance between any two selected sets is at least d. FPT algorithms parameterized by k+𝓁, where 𝓁 = max_{D ∈ 𝒟}|D|, and k+d have been actively studied recently for several specific domains. This paper provides unified algorithmic frameworks to solve this problem. Specifically, for each parameterization k+𝓁 and k+d, we provide an FPT oracle algorithm for the max-min diversification problem using oracles related to 𝒟. We then demonstrate that our frameworks provide the first FPT algorithms on several new domains 𝒟, including the domain of t-linear matroid intersection, almost 2-SAT, minimum edge s,t-flows, vertex sets of s,t-mincut, vertex sets of edge bipartization, and Steiner trees. We also demonstrate that our frameworks generalize most of the existing domain-specific tractability results. Our main technical breakthrough is introducing the notion of max-distance sparsifier of 𝒟, a domain on which the max-min diversification problem is equivalent to the same problem on the original domain 𝒟. The core of our framework is to design FPT oracle algorithms that construct a constant-size max-distance sparsifier of 𝒟. Using max-distance sparsifiers, we provide FPT algorithms for the max-min and max-sum diversification problems on 𝒟, as well as k-center and k-sum-of-radii clustering problems on 𝒟, which are also natural problems in the context of diversification and have their own interests.
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Soh Kumabe. Max-Distance Sparsification for Diversification and Clustering. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kumabe:LIPIcs.ESA.2025.46,
  author =	{Kumabe, Soh},
  title =	{{Max-Distance Sparsification for Diversification and Clustering}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{46:1--46:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.46},
  URN =		{urn:nbn:de:0030-drops-245146},
  doi =		{10.4230/LIPIcs.ESA.2025.46},
  annote =	{Keywords: Fixed-Parameter Tractability, Diversification, Clustering}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.62
Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering

Authors: Sina Bagheri Nezhad, Sayan Bandyapadhyay, and Tianzhi Chen

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
In a seminal work, Chierichetti et al. [Chierichetti et al., 2017] introduced the (t,k)-fair clustering problem: Given a set of red points and a set of blue points in a metric space, a clustering is called fair if the number of red points in each cluster is at most t times and at least 1/t times the number of blue points in that cluster. The goal is to compute a fair clustering with at most k clusters that optimizes certain objective function. Considering this problem, they designed a polynomial-time O(1)- and O(t)-approximation for the k-center and the k-median objective, respectively. Recently, Carta et al. [Carta et al., 2024] studied this problem with the sum-of-radii objective and obtained a (6+ε)-approximation with running time O((k log_{1+ε}(k/ε))^k n^O(1)), i.e., fixed-parameter tractable in k. Here n is the input size. In this work, we design the first polynomial-time O(1)-approximation for (t,k)-fair clustering with the sum-of-radii objective, improving the result of Carta et al. Our result places sum-of-radii in the same group of objectives as k-center, that admit polynomial-time O(1)-approximations. This result also implies a polynomial-time O(1)-approximation for the Euclidean version of the problem, for which an f(k)⋅n^O(1)-time (1+ε)-approximation was known due to Drexler et al. [Drexler et al., 2023]. Here f is an exponential function of k. We are also able to extend our result to any arbitrary 𝓁 ≥ 2 number of colors when t = 1. This matches known results for the k-center and k-median objectives in this case. The significant disparity of sum-of-radii compared to k-center and k-median presents several complex challenges, all of which we successfully overcome in our work. Our main contribution is a novel cluster-merging-based analysis technique for sum-of-radii that helps us achieve the constant-approximation bounds.
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Sina Bagheri Nezhad, Sayan Bandyapadhyay, and Tianzhi Chen. Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bagherinezhad_et_al:LIPIcs.ESA.2025.62,
  author =	{Bagheri Nezhad, Sina and Bandyapadhyay, Sayan and Chen, Tianzhi},
  title =	{{Polynomial-Time Constant-Approximation for Fair Sum-Of-Radii Clustering}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{62:1--62:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.62},
  URN =		{urn:nbn:de:0030-drops-245309},
  doi =		{10.4230/LIPIcs.ESA.2025.62},
  annote =	{Keywords: fair clustering, sum-of-radii clustering, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.67
Compact Representation of Semilinear and Terrain-Like Graphs

Authors: Jean Cardinal and Yelena Yuditsky

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We consider the existence and construction of biclique covers of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The size of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size O(npolylog n). This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz’s problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (Forum Math. Sigma, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size O(nlog³ n). This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (Discrete Comput. Geom. 1994). Finally, we prove that there exists families of unit disk graphs on n vertices that do not admit biclique coverings of size o(n^{4/3}), showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.
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Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2025.67,
  author =	{Cardinal, Jean and Yuditsky, Yelena},
  title =	{{Compact Representation of Semilinear and Terrain-Like Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.67},
  URN =		{urn:nbn:de:0030-drops-245359},
  doi =		{10.4230/LIPIcs.ESA.2025.67},
  annote =	{Keywords: Biclique covers, intersection graphs, visibility graphs, Zarankiewicz’s problem}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.58
Streaming Diameter of High-Dimensional Points

Authors: Magnús M. Halldórsson, Nicolaos Matsakis, and Pavel Veselý

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We improve the space bound for streaming approximation of Diameter but also of Farthest Neighbor queries, Minimum Enclosing Ball and its Coreset, in high-dimensional Euclidean spaces. In particular, our deterministic streaming algorithms store 𝒪(ε^{-2}log(1/(ε))) points. This improves by a factor of ε^{-1} the previous space bound of Agarwal and Sharathkumar (SODA 2010), while retaining the state-of-the-art approximation guarantees, such as √2+ε for Diameter or Farthest Neighbor queries, and also offering a simpler and more complete argument. Moreover, we show that storing Ω(ε^{-1}) points is necessary for a streaming (√2+ε)-approximation of Farthest Pair and Farthest Neighbor queries.
Cite as

Magnús M. Halldórsson, Nicolaos Matsakis, and Pavel Veselý. Streaming Diameter of High-Dimensional Points. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 58:1-58:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{halldorsson_et_al:LIPIcs.ESA.2025.58,
  author =	{Halld\'{o}rsson, Magn\'{u}s M. and Matsakis, Nicolaos and Vesel\'{y}, Pavel},
  title =	{{Streaming Diameter of High-Dimensional Points}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{58:1--58:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.58},
  URN =		{urn:nbn:de:0030-drops-245263},
  doi =		{10.4230/LIPIcs.ESA.2025.58},
  annote =	{Keywords: streaming algorithm, farthest pair, diameter, minimum enclosing ball, coreset}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.25
Covering a Few Submodular Constraints and Applications

Authors: Tanvi Bajpai, Chandra Chekuri, and Pooja Kulkarni

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a cost function c: N → ℝ_+, r monotone submodular functions f_1,f_2,…,f_r over N and requirements b_1,b_2,…,b_r the goal is to find a minimum cost subset S ⊆ N such that f_i(S) ≥ b_i for 1 ≤ i ≤ r. When r = 1 this is the well-known Submodular Set Cover problem. Previous work [Chekuri et al., 2022] considered the setting when r is large and developed bi-criteria approximation algorithms, and approximation algorithms for the important special case when each f_i is a weighted coverage function. These are fairly general models and capture several concrete and interesting problems as special cases. The approximation ratios for these problem are at least Ω(log r) which is unavoidable when r is part of the input. In this paper, motivated by some recent applications, we consider the problem when r is a fixed constant and obtain two main results. When the f_i are weighted coverage functions from a deletion-closed set system we obtain a (1+ε)(e/(e-1))(1+β)-approximation where β is the approximation ratio for the underlying set cover instances via the natural LP. Second, for covering multiple submodular constraints we obtain a randomized bi-criteria approximation algorithm that for any given integer α ≥ 1 outputs a set S such that f_i(S) ≥ (1-1/e^α-ε)b_i for each i ∈ [r] and 𝔼[c(S)] ≤ (1+ε)α ⋅ OPT. These results show that one can obtain nearly as good an approximation for any fixed r as what one would achieve for r = 1. We also demonstrate applications of our results to implicit covering problems such as fair facility location.
Cite as

Tanvi Bajpai, Chandra Chekuri, and Pooja Kulkarni. Covering a Few Submodular Constraints and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bajpai_et_al:LIPIcs.APPROX/RANDOM.2025.25,
  author =	{Bajpai, Tanvi and Chekuri, Chandra and Kulkarni, Pooja},
  title =	{{Covering a Few Submodular Constraints and Applications}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{25:1--25:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.25},
  URN =		{urn:nbn:de:0030-drops-243917},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.25},
  annote =	{Keywords: covering, linear programming, rounding, fairness}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.2
Covering Simple Orthogonal Polygons with Rectangles

Authors: Aniket Basu Roy

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We study the problem of Covering Orthogonal Polygons with Rectangles, focusing on three variants: covering the interior, the boundary, and the corners. While previous work provided constant-factor approximation algorithms for these problems, significant improvements had not been achieved for over two decades. The main contribution of this work is the development of a Polynomial Time Approximation Scheme (PTAS) for both the Boundary Cover and Corner Cover problems on simple polygons, using a local search algorithm. Our work advances the state of the art, improving upon the previous best-known 4-approximation for the Boundary Cover and 2-approximation for the Corner Cover problems. The technical core of our work lies in proving the existence of planar support graphs for certain geometric hypergraphs defined by the polygon and its containment-maximal rectangles. This structural insight enables the application of the local search framework to achieve the PTAS results. We also demonstrate the limitations of this approach by constructing instances where local search fails for the Interior Cover and certain dual problems, such as the Maximum Antirectangle and Hitting Set problems. Additionally, the methods yield a PTAS for a special case of the Discrete Independent Set problem for rectangles. These results not only settle longstanding open questions but also introduce new techniques that may be of independent interest within computational geometry.
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Aniket Basu Roy. Covering Simple Orthogonal Polygons with Rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{basuroy:LIPIcs.APPROX/RANDOM.2025.2,
  author =	{Basu Roy, Aniket},
  title =	{{Covering Simple Orthogonal Polygons with Rectangles}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{2:1--2:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.2},
  URN =		{urn:nbn:de:0030-drops-243686},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.2},
  annote =	{Keywords: Polygon Covering, Approximation Algorithms, Orthogonal Polygons, Rectangles, Local Search, Planar Supports}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.8
Multipass Linear Sketches for Geometric LP-Type Problems

Authors: N. Efe Çekirge, William Gay, and David P. Woodruff

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
LP-type problems such as the Minimum Enclosing Ball (MEB), Linear Support Vector Machine (SVM), Linear Programming (LP), and Semidefinite Programming (SDP) are fundamental combinatorial optimization problems, with many important applications in machine learning applications such as classification, bioinformatics, and noisy learning. We study LP-type problems in several streaming and distributed big data models, giving ε-approximation linear sketching algorithms with a focus on the high accuracy regime with low dimensionality d, that is, when d < (1/ε)^0.999. Our main result is an O(ds) pass algorithm with O(s(√d/ε)^{3d/s}) ⋅ poly(d, log (1/ε)) space complexity in words, for any parameter s ∈ [1, d log (1/ε)], to solve ε-approximate LP-type problems of O(d) combinatorial and VC dimension. Notably, by taking s = d log (1/ε), we achieve space complexity polynomial in d and polylogarithmic in 1/ε, presenting exponential improvements in 1/ε over current algorithms. We complement our results by showing lower bounds of (1/ε)^Ω(d) for any 1-pass algorithm solving the (1 + ε)-approximation MEB and linear SVM problems, further motivating our multi-pass approach.
Cite as

N. Efe Çekirge, William Gay, and David P. Woodruff. Multipass Linear Sketches for Geometric LP-Type Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 8:1-8:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cekirge_et_al:LIPIcs.APPROX/RANDOM.2025.8,
  author =	{\c{C}ekirge, N. Efe and Gay, William and Woodruff, David P.},
  title =	{{Multipass Linear Sketches for Geometric LP-Type Problems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{8:1--8:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.8},
  URN =		{urn:nbn:de:0030-drops-243741},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.8},
  annote =	{Keywords: Streaming, sketching, LP-type problems}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.12
Relational Approximations for Subspace Primitives

Authors: Xiang Liu and Kasturi Varadarajan

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We explore fundamental geometric computations on point sets that are given to the algorithm implicitly. In particular, we are given a database which is a collection of tables with numerical values, and the geometric computation is to be performed on the join of the tables. Explicitly computing this join takes time exponential in the size of the tables. We are therefore interested in geometric problems that can be solved by algorithms whose running time is a polynomial in the size of the tables. Such relational algorithms are typically not able to explicitly compute the join. To avoid the NP-completeness bottleneck, researchers assume that the tables have a tractable combinatorial structure, like being acyclic. Even with this assumption, simple geometric computations turn out to be non-trivial and sometimes intractable. In this article, we study the problem of computing the maximum distance of a point in the join to a given subspace, and develop approximation algorithms for this NP-hard problem.
Cite as

Xiang Liu and Kasturi Varadarajan. Relational Approximations for Subspace Primitives. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{liu_et_al:LIPIcs.APPROX/RANDOM.2025.12,
  author =	{Liu, Xiang and Varadarajan, Kasturi},
  title =	{{Relational Approximations for Subspace Primitives}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{12:1--12:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.12},
  URN =		{urn:nbn:de:0030-drops-243781},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.12},
  annote =	{Keywords: relational algorithm, Euclidean distance, subspace approximation}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.23
Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints

Authors: Sayan Bandyapadhyay and Tianzhi Chen

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
In this work, we study k-min-sum-of-radii (k-MSR) clustering under mergeable constraints. k-MSR seeks to group data points using a set of up to k balls, such that the sum of the radii of the balls is minimized. A clustering constraint is called mergeable if merging two clusters satisfying the constraint, results in a cluster that also satisfies the constraint. Many popularly studied constraints are mergeable, including fairness constraints and lower bound constraints. In our work, we design a (4+ε)-approximation for k-MSR under any given mergeable constraint with runtime 2^{O(k/(ε)⋅log²k/ε)} n⁴, i.e., fixed-parameter tractable in k for constant ε. Our result directly improves upon the FPT (6+ε)-approximation by Carta et al. [Carta et al., 2024]. We also provide a hardness result that excludes the exact solvability of k-MSR under any given mergeable constraint in time f(k)n^o(k), assuming ETH is true.
Cite as

Sayan Bandyapadhyay and Tianzhi Chen. Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.APPROX/RANDOM.2025.23,
  author =	{Bandyapadhyay, Sayan and Chen, Tianzhi},
  title =	{{Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{23:1--23:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.23},
  URN =		{urn:nbn:de:0030-drops-243894},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.23},
  annote =	{Keywords: sum-of-radii clustering, mergeable constraints, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.7
Approximation and Parameterized Algorithms for Covering with Disks of Two Types of Radii

Authors: Sayan Bandyapadhyay and Eli Mitchell

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We study the Discrete Covering with Two Types of Radii problem motivated by its application in wireless networks. In this problem, the goal is to assign either small-range high frequency or large-range low frequency to each access point, maximizing the number of users in high-frequency regions while ensuring that each user is in the range of an access point. Unlike other weighted covering problems, our problem requires satisfying two simultaneous objectives, which calls for novel approaches that leverage the underlying geometry of the problem. In our work, we present two new algorithms: the first is a polynomial-time (2.5 + ε)-approximation, and the second is an exact algorithm for sparse instances, which is fixed-parameter tractable (FPT) in the number of large-radius disks. We also prove that such an FPT algorithm is impossible for general instances lacking sparsity, assuming the Exponential Time Hypothesis. Before our work, the best-known polynomial-time approximation factor was 4 for the problem. Our approximation algorithm results from a fine-grained classification of points that can contribute to the gain of a solution. Based on this classification, we design two sub-algorithms with interdependent guarantees to recover the respective class of points as gain. Our algorithm exploits further properties of Delaunay triangulations to achieve the improved bound. The FPT algorithm is based on branching that utilizes the sparsity of the instances to limit the overall search space.
Cite as

Sayan Bandyapadhyay and Eli Mitchell. Approximation and Parameterized Algorithms for Covering with Disks of Two Types of Radii. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.WADS.2025.7,
  author =	{Bandyapadhyay, Sayan and Mitchell, Eli},
  title =	{{Approximation and Parameterized Algorithms for Covering with Disks of Two Types of Radii}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.7},
  URN =		{urn:nbn:de:0030-drops-242386},
  doi =		{10.4230/LIPIcs.WADS.2025.7},
  annote =	{Keywords: Covering, Disks, Approximation, FPT}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.61
How to Compute the Volume in Low Dimension?

Authors: Arjan Cornelissen, Simon Apers, and Sander Gribling

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.
Cite as

Arjan Cornelissen, Simon Apers, and Sander Gribling. How to Compute the Volume in Low Dimension?. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 61:1-61:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cornelissen_et_al:LIPIcs.ICALP.2025.61,
  author =	{Cornelissen, Arjan and Apers, Simon and Gribling, Sander},
  title =	{{How to Compute the Volume in Low Dimension?}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{61:1--61:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.61},
  URN =		{urn:nbn:de:0030-drops-234381},
  doi =		{10.4230/LIPIcs.ICALP.2025.61},
  annote =	{Keywords: Query complexity, computational geometry, quantum computing, volume estimation, high-precision limit}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.62
On Approximability of 𝓁₂² Min-Sum Clustering

Authors: Karthik C. S., Euiwoong Lee, Yuval Rabani, Chris Schwiegelshohn, and Samson Zhou

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0.
Cite as

Karthik C. S., Euiwoong Lee, Yuval Rabani, Chris Schwiegelshohn, and Samson Zhou. On Approximability of 𝓁₂² Min-Sum Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{karthikc.s._et_al:LIPIcs.SoCG.2025.62,
  author =	{Karthik C. S. and Lee, Euiwoong and Rabani, Yuval and Schwiegelshohn, Chris and Zhou, Samson},
  title =	{{On Approximability of 𝓁₂² Min-Sum Clustering}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{62:1--62:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.62},
  URN =		{urn:nbn:de:0030-drops-232142},
  doi =		{10.4230/LIPIcs.SoCG.2025.62},
  annote =	{Keywords: Clustering, hardness of approximation, polynomial-time approximation schemes, learning-augmented algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.20
Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems

Authors: Ahmad Biniaz, Anil Maheshwari, Magnus Christian Ring Merrild, Joseph S. B. Mitchell, Saeed Odak, Valentin Polishchuk, Eliot W. Robson, Casper Moldrup Rysgaard, Jens Kristian Refsgaard Schou, Thomas Shermer, Jack Spalding-Jamieson, Rolf Svenning, and Da Wei Zheng

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We introduce the contiguous art gallery problem which is to guard the boundary of a simple polygon with a minimum number of guards such that each guard covers exactly one contiguous portion of the boundary. Art gallery problems are often NP-hard. In particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguity constraint. This paper is a merge of three concurrent works [Ahmad Biniaz et al., 2024; Magnus Christian Ring Merrild et al., 2024; Eliot W. Robson et al., 2024] each showing that (surprisingly) the contiguous art gallery problem is solvable in polynomial time. The common idea of all three approaches is developing a greedy function that maps a point on the boundary to the furthest point on the boundary so that the contiguous interval along the boundary between them could be guarded by one guard. Repeatedly applying this function immediately leads to an OPT+1 approximation. By studying this greedy algorithm, we present three different approaches that achieve an optimal solution. The first and second approach apply this greedy algorithm from different points on the boundary that could be found in advance or on the fly while traversing along the boundary (respectively). The third approach represents this function as a piecewise linear rational function, which can be reduced to an abstract arc cover problem involving infinite families of arcs. We identify other problems that can be represented by similar functions, and solve them via the third approach. From the combinatorial point of view, we show that any n-vertex polygon can be guarded by at most ⌊(n-2)/2⌋ guards. This bound is tight because there are polygons that require this many guards.
Cite as

Ahmad Biniaz, Anil Maheshwari, Magnus Christian Ring Merrild, Joseph S. B. Mitchell, Saeed Odak, Valentin Polishchuk, Eliot W. Robson, Casper Moldrup Rysgaard, Jens Kristian Refsgaard Schou, Thomas Shermer, Jack Spalding-Jamieson, Rolf Svenning, and Da Wei Zheng. Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2025.20,
  author =	{Biniaz, Ahmad and Maheshwari, Anil and Merrild, Magnus Christian Ring and Mitchell, Joseph S. B. and Odak, Saeed and Polishchuk, Valentin and Robson, Eliot W. and Rysgaard, Casper Moldrup and Schou, Jens Kristian Refsgaard and Shermer, Thomas and Spalding-Jamieson, Jack and Svenning, Rolf and Zheng, Da Wei},
  title =	{{Polynomial-Time Algorithms for Contiguous Art Gallery and Related Problems}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{20:1--20:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.20},
  URN =		{urn:nbn:de:0030-drops-231720},
  doi =		{10.4230/LIPIcs.SoCG.2025.20},
  annote =	{Keywords: Art Gallery Problem, Computational Geometry, Combinatorics, Discrete Algorithms}
}
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