10 Search Results for "Awasthi, Pranjal"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.33
Near-Optimal Bounds for Parameterized Euclidean k-Means

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1). In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk). Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Near-Optimal Bounds for Parameterized Euclidean k-Means. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.33,
  author =	{Cohen-Addad, Vincent and C. S., Karthik and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Near-Optimal Bounds for Parameterized Euclidean k-Means}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.33},
  URN =		{urn:nbn:de:0030-drops-258391},
  doi =		{10.4230/LIPIcs.SoCG.2026.33},
  annote =	{Keywords: k-means clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.34
Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points). We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.34,
  author =	{Cohen-Addad, Vincent and Karthik C. S. and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.34},
  URN =		{urn:nbn:de:0030-drops-258404},
  doi =		{10.4230/LIPIcs.SoCG.2026.34},
  annote =	{Keywords: k-means clustering, k-median clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.48
FPT Approximations for Capacitated Sum of Radii and Diameters

Authors: Arnold Filtser and Ameet Gadekar

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The Capacitated Sum of Radii problem involves partitioning a set of points P, where each point p ∈ P has capacity U_p, into k clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point p is at most U_p. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a ≈5.83-approximation algorithm in FPT time (improving a previous ≈7.61 approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.
Cite as

Arnold Filtser and Ameet Gadekar. FPT Approximations for Capacitated Sum of Radii and Diameters. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{filtser_et_al:LIPIcs.SoCG.2026.48,
  author =	{Filtser, Arnold and Gadekar, Ameet},
  title =	{{FPT Approximations for Capacitated Sum of Radii and Diameters}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{48:1--48:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.48},
  URN =		{urn:nbn:de:0030-drops-258545},
  doi =		{10.4230/LIPIcs.SoCG.2026.48},
  annote =	{Keywords: clustering, sum of radii, sum of diameter, capacitated clustering, fpt}
}
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.
Cite as

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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.29
Guessing Efficiently for Constrained Subspace Approximation

Authors: Aditya Bhaskara, Sepideh Mahabadi, Madhusudhan Reddy Pittu, Ali Vakilian, and David P. Woodruff

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

Abstract
In this paper we study constrained subspace approximation problem. Given a set of n points {a₁,…,a_n} in ℝ^d, the goal of the subspace approximation problem is to find a k dimensional subspace that best approximates the input points. More precisely, for a given p ≥ 1, we aim to minimize the pth power of the 𝓁_p norm of the error vector (‖a₁-Pa₁‖,…,‖a_n-Pa_n‖), where P denotes the projection matrix onto the subspace and the norms are Euclidean. In constrained subspace approximation (CSA), we additionally have constraints on the projection matrix P. In its most general form, we require P to belong to a given subset 𝒮 that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either (1+ε)-multiplicative or ε-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to fair subspace approximation, k-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for k-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.
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Aditya Bhaskara, Sepideh Mahabadi, Madhusudhan Reddy Pittu, Ali Vakilian, and David P. Woodruff. Guessing Efficiently for Constrained Subspace Approximation. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhaskara_et_al:LIPIcs.ICALP.2025.29,
  author =	{Bhaskara, Aditya and Mahabadi, Sepideh and Pittu, Madhusudhan Reddy and Vakilian, Ali and Woodruff, David P.},
  title =	{{Guessing Efficiently for Constrained Subspace Approximation}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{29:1--29:20},
  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.2025.29},
  URN =		{urn:nbn:de:0030-drops-234068},
  doi =		{10.4230/LIPIcs.ICALP.2025.29},
  annote =	{Keywords: parameterized complexity, low rank approximation, fairness, non-negative matrix factorization, clustering}
}
Document
DOI: 10.4230/LIPIcs.FORC.2023.10
Distributionally Robust Data Join

Authors: Pranjal Awasthi, Christopher Jung, and Jamie Morgenstern

Published in: LIPIcs, Volume 256, 4th Symposium on Foundations of Responsible Computing (FORC 2023)

Abstract
Suppose we are given two datasets: a labeled dataset and unlabeled dataset which also has additional auxiliary features not present in the first dataset. What is the most principled way to use these datasets together to construct a predictor? The answer should depend upon whether these datasets are generated by the same or different distributions over their mutual feature sets, and how similar the test distribution will be to either of those distributions. In many applications, the two datasets will likely follow different distributions, but both may be close to the test distribution. We introduce the problem of building a predictor which minimizes the maximum loss over all probability distributions over the original features, auxiliary features, and binary labels, whose Wasserstein distance is r₁ away from the empirical distribution over the labeled dataset and r₂ away from that of the unlabeled dataset. This can be thought of as a generalization of distributionally robust optimization (DRO), which allows for two data sources, one of which is unlabeled and may contain auxiliary features.
Cite as

Pranjal Awasthi, Christopher Jung, and Jamie Morgenstern. Distributionally Robust Data Join. In 4th Symposium on Foundations of Responsible Computing (FORC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 256, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{awasthi_et_al:LIPIcs.FORC.2023.10,
  author =	{Awasthi, Pranjal and Jung, Christopher and Morgenstern, Jamie},
  title =	{{Distributionally Robust Data Join}},
  booktitle =	{4th Symposium on Foundations of Responsible Computing (FORC 2023)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-272-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{256},
  editor =	{Talwar, Kunal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2023.10},
  URN =		{urn:nbn:de:0030-drops-179311},
  doi =		{10.4230/LIPIcs.FORC.2023.10},
  annote =	{Keywords: Distributionally Robust Optimization, Semi-Supervised Learning, Learning Theory}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.4
Hardness of Approximation for Euclidean k-Median

Authors: Anup Bhattacharya, Dishant Goyal, and Ragesh Jaiswal

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
The Euclidean k-median problem is defined in the following manner: given a set 𝒳 of n points in d-dimensional Euclidean space ℝ^d, and an integer k, find a set C ⊂ ℝ^d of k points (called centers) such that the cost function Φ(C,𝒳) ≡ ∑_{x ∈ 𝒳} min_{c ∈ C} ‖x-c‖₂ is minimized. The Euclidean k-means problem is defined similarly by replacing the distance with squared Euclidean distance in the cost function. Various hardness of approximation results are known for the Euclidean k-means problem [Pranjal Awasthi et al., 2015; Euiwoong Lee et al., 2017; Vincent Cohen{-}Addad and {Karthik {C. S.}}, 2019]. However, no hardness of approximation result was known for the Euclidean k-median problem. In this work, assuming the unique games conjecture (UGC), we provide the hardness of approximation result for the Euclidean k-median problem in O(log k) dimensional space. This solves an open question posed explicitly in the work of Awasthi et al. [Pranjal Awasthi et al., 2015]. Furthermore, we study the hardness of approximation for the Euclidean k-means/k-median problems in the bi-criteria setting where an algorithm is allowed to choose more than k centers. That is, bi-criteria approximation algorithms are allowed to output β k centers (for constant β > 1) and the approximation ratio is computed with respect to the optimal k-means/k-median cost. We show the hardness of bi-criteria approximation result for the Euclidean k-median problem for any β < 1.015, assuming UGC. We also show a similar hardness of bi-criteria approximation result for the Euclidean k-means problem with a stronger bound of β < 1.28, again assuming UGC.
Cite as

Anup Bhattacharya, Dishant Goyal, and Ragesh Jaiswal. Hardness of Approximation for Euclidean k-Median. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhattacharya_et_al:LIPIcs.APPROX/RANDOM.2021.4,
  author =	{Bhattacharya, Anup and Goyal, Dishant and Jaiswal, Ragesh},
  title =	{{Hardness of Approximation for Euclidean k-Median}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{4:1--4:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.4},
  URN =		{urn:nbn:de:0030-drops-146979},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.4},
  annote =	{Keywords: Hardness of approximation, bicriteria approximation, approximation algorithms, k-median, k-means}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.7
Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem

Authors: Haris Angelidakis, Pranjal Awasthi, Avrim Blum, Vaggos Chatziafratis, and Chen Dan

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

Abstract
We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is gamma-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most gamma >= 1. The goal then is to efficiently recover this "pronounced" optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving O~(Delta/sqrt(log Delta))-stable instances on graphs of maximum degree Delta, (k - 1)-stable instances on k-colorable graphs and (1 + epsilon)-stable instances on planar graphs (for any fixed epsilon > 0), using both combinatorial techniques as well as LPs and the Sherali-Adams hierarchy. For general graphs, we present a strong lower bound showing that there are no efficient algorithms for O(n^(1/2 - epsilon))-stable instances of Independent Set, assuming the planted clique conjecture. To complement our negative result, we give an algorithm for (epsilon n)-stable instances, for any fixed epsilon > 0. As a by-product of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a gamma-certified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of gamma-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of gamma >= 1 (hence, such algorithms not only solve gamma-stable instances optimally, but also have guarantees even on unstable instances). Here, we obtain Delta-certified algorithms for Independent Set on graphs of maximum degree Delta, and (1+epsilon)-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((Delta + 1)/3 + epsilon)-certified algorithm for Independent Set on graphs of maximum degree Delta where all weights are equal to 1.
Cite as

Haris Angelidakis, Pranjal Awasthi, Avrim Blum, Vaggos Chatziafratis, and Chen Dan. Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{angelidakis_et_al:LIPIcs.ESA.2019.7,
  author =	{Angelidakis, Haris and Awasthi, Pranjal and Blum, Avrim and Chatziafratis, Vaggos and Dan, Chen},
  title =	{{Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{7:1--7:16},
  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.2019.7},
  URN =		{urn:nbn:de:0030-drops-111288},
  doi =		{10.4230/LIPIcs.ESA.2019.7},
  annote =	{Keywords: Bilu-Linial stability, perturbation resilience, beyond worst-case analysis, Independent Set, Vertex Cover, Multiway Cut}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.18
Robust Communication-Optimal Distributed Clustering Algorithms

Authors: Pranjal Awasthi, Ainesh Bakshi, Maria-Florina Balcan, Colin White, and David P. Woodruff

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
In this work, we study the k-median and k-means clustering problems when the data is distributed across many servers and can contain outliers. While there has been a lot of work on these problems for worst-case instances, we focus on gaining a finer understanding through the lens of beyond worst-case analysis. Our main motivation is the following: for many applications such as clustering proteins by function or clustering communities in a social network, there is some unknown target clustering, and the hope is that running a k-median or k-means algorithm will produce clusterings which are close to matching the target clustering. Worst-case results can guarantee constant factor approximations to the optimal k-median or k-means objective value, but not closeness to the target clustering. Our first result is a distributed algorithm which returns a near-optimal clustering assuming a natural notion of stability, namely, approximation stability [Awasthi and Balcan, 2014], even when a constant fraction of the data are outliers. The communication complexity is O~(sk+z) where s is the number of machines, k is the number of clusters, and z is the number of outliers. Next, we show this amount of communication cannot be improved even in the setting when the input satisfies various non-worst-case assumptions. We give a matching Omega(sk+z) lower bound on the communication required both for approximating the optimal k-means or k-median cost up to any constant, and for returning a clustering that is close to the target clustering in Hamming distance. These lower bounds hold even when the data satisfies approximation stability or other common notions of stability, and the cluster sizes are balanced. Therefore, Omega(sk+z) is a communication bottleneck, even for real-world instances.
Cite as

Pranjal Awasthi, Ainesh Bakshi, Maria-Florina Balcan, Colin White, and David P. Woodruff. Robust Communication-Optimal Distributed Clustering Algorithms. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{awasthi_et_al:LIPIcs.ICALP.2019.18,
  author =	{Awasthi, Pranjal and Bakshi, Ainesh and Balcan, Maria-Florina and White, Colin and Woodruff, David P.},
  title =	{{Robust Communication-Optimal Distributed Clustering Algorithms}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.18},
  URN =		{urn:nbn:de:0030-drops-105942},
  doi =		{10.4230/LIPIcs.ICALP.2019.18},
  annote =	{Keywords: robust distributed clustering, communication complexity}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.754
The Hardness of Approximation of Euclidean k-Means

Authors: Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d. In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.
Cite as

Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 754-767, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{awasthi_et_al:LIPIcs.SOCG.2015.754,
  author =	{Awasthi, Pranjal and Charikar, Moses and Krishnaswamy, Ravishankar and Sinop, Ali Kemal},
  title =	{{The Hardness of Approximation of Euclidean k-Means}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{754--767},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.754},
  URN =		{urn:nbn:de:0030-drops-51178},
  doi =		{10.4230/LIPIcs.SOCG.2015.754},
  annote =	{Keywords: Euclidean k-means, Hardness of Approximation, Vertex Cover}
}
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