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

Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric:
* For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost.
* For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point.
* An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2.

Pankaj K. Agarwal, Kyle Fox, Debmalya Panigrahi, Kasturi R. Varadarajan, and Allen Xiao. Faster Algorithms for the Geometric Transportation Problem. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2017.7, author = {Agarwal, Pankaj K. and Fox, Kyle and Panigrahi, Debmalya and Varadarajan, Kasturi R. and Xiao, Allen}, title = {{Faster Algorithms for the Geometric Transportation Problem}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {7:1--7:16}, 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.2017.7}, URN = {urn:nbn:de:0030-drops-72344}, doi = {10.4230/LIPIcs.SoCG.2017.7}, annote = {Keywords: transportation map, earth mover's distance, shape matching, approximation algorithms} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

In this work, we address the following question. Suppose we are given a set D of positive-weighted disks and a set T of n points in the plane, such that each point of T is contained in at least two disks of D. Then is there always a subset S of D such that the union of the disks in S contains all the points of T and the total weight of the disks of D that are not in S is at least a constant fraction of the total weight of the disks in D?
In our work, we prove the Extraction Theorem that answers this question in the affirmative. Our constructive proof heavily exploits the geometry of disks, and in the process, we make interesting connections between our work and the literature on local search for geometric optimization problems.
The Extraction Theorem helps to design the first polynomial-time O(1)-approximations for two important geometric covering problems involving disks.

Sayan Bandyapadhyay, Anil Maheshwari, Sasanka Roy, Michiel Smid, and Kasturi Varadarajan. Geometric Covering via Extraction Theorem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ITCS.2024.7, author = {Bandyapadhyay, Sayan and Maheshwari, Anil and Roy, Sasanka and Smid, Michiel and Varadarajan, Kasturi}, title = {{Geometric Covering via Extraction Theorem}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {7:1--7:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.7}, URN = {urn:nbn:de:0030-drops-195355}, doi = {10.4230/LIPIcs.ITCS.2024.7}, annote = {Keywords: Covering, Extraction theorem, Double-disks, Submodularity, Local search} }

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

In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k_i balls of radius r_i, for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded, assuming 𝖯 ≠ NP. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture - currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of k-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.

Tanmay Inamdar and Kasturi Varadarajan. Non-Uniform k-Center and Greedy Clustering. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{inamdar_et_al:LIPIcs.SWAT.2022.28, author = {Inamdar, Tanmay and Varadarajan, Kasturi}, title = {{Non-Uniform k-Center and Greedy Clustering}}, booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)}, pages = {28:1--28:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-236-5}, ISSN = {1868-8969}, year = {2022}, volume = {227}, editor = {Czumaj, Artur and Xin, Qin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.28}, URN = {urn:nbn:de:0030-drops-161881}, doi = {10.4230/LIPIcs.SWAT.2022.28}, annote = {Keywords: k-center, approximation algorithms, non-uniform k-center, clustering} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

In sum of radii clustering, the input consists of a finite set of points in a metric space. The problem asks to place a set of k balls centered at a subset of the points such that every point is covered by some ball, and the objective is to minimize the sum of radii of these balls. In the capacitated version of the problem, we want to assign each point to a ball containing it, such that no ball is assigned more than U points, where U denotes the capacity of the points. While constant approximations are known for the uncapacitated version of the problem, there is no work on the capacitated version. We make progress on this problem by obtaining a constant approximation using a Fixed Parameter Tractable (FPT) algorithm. In particular, the running time of the algorithm is of the form 2^O(k²) ⋅ n^O(1). As a warm-up for this result, we also give a constant approximation for the uncapacitated sum of radii clustering problem with matroid constraints, thus obtaining the first FPT approximation for this problem.

Tanmay Inamdar and Kasturi Varadarajan. Capacitated Sum-Of-Radii Clustering: An FPT Approximation. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 62:1-62:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{inamdar_et_al:LIPIcs.ESA.2020.62, author = {Inamdar, Tanmay and Varadarajan, Kasturi}, title = {{Capacitated Sum-Of-Radii Clustering: An FPT Approximation}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {62:1--62:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.62}, URN = {urn:nbn:de:0030-drops-129288}, doi = {10.4230/LIPIcs.ESA.2020.62}, annote = {Keywords: Sum-of-radii Clustering, Capacitated Clustering} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

In this paper, we consider the colorful k-center problem, which is a generalization of the well-known k-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius rho, such that with k balls of radius rho, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs.

Sayan Bandyapadhyay, Tanmay Inamdar, Shreyas Pai, and Kasturi Varadarajan. A Constant Approximation for Colorful k-Center. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ESA.2019.12, author = {Bandyapadhyay, Sayan and Inamdar, Tanmay and Pai, Shreyas and Varadarajan, Kasturi}, title = {{A Constant Approximation for Colorful k-Center}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.12}, URN = {urn:nbn:de:0030-drops-111336}, doi = {10.4230/LIPIcs.ESA.2019.12}, annote = {Keywords: Colorful k-center, Euclidean Plane, LP Rounding, Outliers} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

In the minimum constraint removal problem, we are given a set of geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable, and (perhaps surprisingly) no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is a new approximation technique that gives O(sqrt{n})-approximation for rectangles, disks as well as rectilinear polygons. The technique also gives O(sqrt{n})-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.

Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi Varadarajan. Improved Approximation Bounds for the Minimum Constraint Removal Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.APPROX-RANDOM.2018.2, author = {Bandyapadhyay, Sayan and Kumar, Neeraj and Suri, Subhash and Varadarajan, Kasturi}, title = {{Improved Approximation Bounds for the Minimum Constraint Removal Problem}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {2:1--2:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.2}, URN = {urn:nbn:de:0030-drops-94066}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.2}, annote = {Keywords: Minimum Constraint Removal, Minimum Color Path, Barrier Resilience, Obstacle Removal, Obstacle Free Path, Approximation} }

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

In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B' subseteq B of balls and assign each point in P to some ball in B' that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B'.
We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1+epsilon) factor expansion is sufficient for any epsilon > 0, with the approximation factor being a polynomial in 1/epsilon. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.

Sayan Bandyapadhyay, Santanu Bhowmick, Tanmay Inamdar, and Kasturi Varadarajan. Capacitated Covering Problems in Geometric Spaces. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2018.7, author = {Bandyapadhyay, Sayan and Bhowmick, Santanu and Inamdar, Tanmay and Varadarajan, Kasturi}, title = {{Capacitated Covering Problems in Geometric Spaces}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.7}, URN = {urn:nbn:de:0030-drops-87205}, doi = {10.4230/LIPIcs.SoCG.2018.7}, annote = {Keywords: Capacitated covering, Geometric set cover, LP rounding, Bi-criteria approximation} }

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

We study a generalization of the Set Cover problem called the Partial Set Cover in the context of geometric set systems. The input to this problem is a set system (X, R), where X is a set of elements and R is a collection of subsets of X, and an integer k <= |X|. Each set in R has a non-negative weight associated with it. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from R. The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for a certain projection of the set system (X, R). As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems.

Tanmay Inamdar and Kasturi Varadarajan. On Partial Covering For Geometric Set Systems. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{inamdar_et_al:LIPIcs.SoCG.2018.47, author = {Inamdar, Tanmay and Varadarajan, Kasturi}, title = {{On Partial Covering For Geometric Set Systems}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.47}, URN = {urn:nbn:de:0030-drops-87607}, doi = {10.4230/LIPIcs.SoCG.2018.47}, annote = {Keywords: Partial Set Cover, Geometric Set Cover} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

In this paper we consider two metric covering/clustering problems - Minimum Cost Covering Problem (MCC) and k-clustering. In the MCC problem, we are given two point sets X (clients) and Y (servers), and a metric on X cup Y. We would like to cover the clients by balls centered at the servers. The objective function to minimize is the sum of the alpha-th power of the radii of the balls. Here alpha geq 1 is a parameter of the problem (but not of a problem instance). MCC is closely related to the k-clustering problem. The main difference between k-clustering and MCC is that in k-clustering one needs to select k balls to cover the clients.
For any eps > 0, we describe quasi-polynomial time (1 + eps) approximation algorithms for both of the problems. However, in case of k-clustering the algorithm uses (1 + eps)k balls. Prior to our work, a 3^alpha and a c^alpha approximation were achieved by polynomial-time algorithms for MCC and k-clustering, respectively, where c > 1 is an absolute constant. These two problems are thus interesting examples of metric covering/clustering problems that admit (1 + eps)-approximation (using (1 + eps)k balls in case of k-clustering), if one is willing to settle for quasi-polynomial time. In contrast, for the variant of MCC where alpha is part of the input, we show under standard assumptions that no polynomial time algorithm can achieve an approximation factor better than O(log |X|) for alpha geq log |X|.

Sayan Bandyapadhyay and Kasturi Varadarajan. Approximate Clustering via Metric Partitioning. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ISAAC.2016.15, author = {Bandyapadhyay, Sayan and Varadarajan, Kasturi}, title = {{Approximate Clustering via Metric Partitioning}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.15}, URN = {urn:nbn:de:0030-drops-67751}, doi = {10.4230/LIPIcs.ISAAC.2016.15}, annote = {Keywords: Approximation Algorithms, Clustering, Covering, Probabilistic Parti- tions} }

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

Clustering problems often arise in fields like data mining and machine learning. Clustering usually refers to the task of partitioning a collection of objects into groups with similar elements, with respect to a similarity (or dissimilarity) measure. Among the clustering problems, k-means clustering in particular has received much attention from researchers. Despite the fact that k-means is a well studied problem, its status in the plane is still open. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound achievable in polynomial time is 9+epsilon.
In this paper, we consider the following variant of k-means. Given a set C of points in R^d and a real f > 0, find a finite set F of points in R^d that minimizes the quantity f*|F|+sum_{p in C} min_{q in F} {||p-q||}^2. For any fixed dimension d, we design a PTAS for this problem that is based on local search. We also give a "bi-criterion" local search algorithm for k-means which uses (1+epsilon)k centers and yields a solution whose cost is at most (1+epsilon) times the cost of an optimal k-means solution. The algorithm runs in polynomial time for any fixed dimension.
The contribution of this paper is two-fold. On the one hand, we are able to handle the square of distances in an elegant manner, obtaining a near-optimal approximation bound. This leads us towards a better understanding of the k-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that little is known about the local search method for geometric approximation.

Sayan Bandyapadhyay and Kasturi Varadarajan. On Variants of k-means Clustering. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2016.14, author = {Bandyapadhyay, Sayan and Varadarajan, Kasturi}, title = {{On Variants of k-means Clustering}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {14:1--14:15}, 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.2016.14}, URN = {urn:nbn:de:0030-drops-59061}, doi = {10.4230/LIPIcs.SoCG.2016.14}, annote = {Keywords: k-means, Facility location, Local search, Geometric approximation} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

In this article, we study shape fitting problems, epsilon-coresets, and total sensitivity. We focus on the (j,k)-projective clustering problems, including k-median/k-means, k-line clustering, j-subspace approximation, and the integer (j,k)-projective clustering problem. We derive upper bounds of total sensitivities for these problems, and obtain epsilon-coresets using these upper bounds. Using a dimension-reduction type argument, we are able to greatly simplify earlier results on total sensitivity for the k-median/k-means clustering problems, and obtain positively-weighted epsilon-coresets for several variants of the (j,k)-projective clustering problem. We also extend an earlier result on epsilon-coresets for the integer (j,k)-projective clustering problem in fixed dimension to the case of high dimension.

Kasturi Varadarajan and Xin Xiao. On the Sensitivity of Shape Fitting Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 486-497, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{varadarajan_et_al:LIPIcs.FSTTCS.2012.486, author = {Varadarajan, Kasturi and Xiao, Xin}, title = {{On the Sensitivity of Shape Fitting Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {486--497}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.486}, URN = {urn:nbn:de:0030-drops-38830}, doi = {10.4230/LIPIcs.FSTTCS.2012.486}, annote = {Keywords: Coresets, shape fitting, k-means, subspace approximation} }

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