20 Search Results for "Bhattacharya, Anup"

Document
DOI: 10.4230/LIPIcs.SEA.2026.26
Engineering Algorithms for Dynamic Greedy Set Cover

Authors: Amitai Uzrad

Published in: LIPIcs, Volume 371, 24th International Symposium on Experimental Algorithms (SEA 2026)

Abstract
In the dynamic set cover problem, the input is a dynamic universe of elements and a fixed collection of sets. As elements are inserted or deleted, the goal is to efficiently maintain an approximate minimum set cover. While the past decade has seen significant theoretical breakthroughs for this problem, a notable gap remains between theoretical design and practical performance, as no comprehensive experimental study currently exists to validate these results. In this paper, we bridge this gap by implementing and evaluating four greedy-based dynamic algorithms across a diverse range of real-world instances. We derive our implementations from state-of-the-art frameworks - such as [GKKP(STOC'17); SU(STOC'23); SUZ(FOCS'24)] - which we simplify by identifying and modifying intricate subroutines that optimize asymptotic bounds but hinder practical performance. We evaluate these algorithms based on solution quality (set cover size) and efficiency, which comprises update time - the time required to update the solution following each insertion/deletion - and recourse - the number of changes made to the solution per update. Each algorithm uses a parameter β to balance quality against efficiency; we investigate the influence of this tradeoff parameter on each algorithm and then perform a comparative analysis to evaluate the algorithms against each other. Our results provide the first practical insights into which algorithmic strategies provide the most value in realistic scenarios.
Cite as

Amitai Uzrad. Engineering Algorithms for Dynamic Greedy Set Cover. In 24th International Symposium on Experimental Algorithms (SEA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 371, pp. 26:1-26:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{uzrad:LIPIcs.SEA.2026.26,
  author =	{Uzrad, Amitai},
  title =	{{Engineering Algorithms for Dynamic Greedy Set Cover}},
  booktitle =	{24th International Symposium on Experimental Algorithms (SEA 2026)},
  pages =	{26:1--26:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-422-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{371},
  editor =	{Aum\"{u}ller, Martin and Finocchi, Irene},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2026.26},
  URN =		{urn:nbn:de:0030-drops-260308},
  doi =		{10.4230/LIPIcs.SEA.2026.26},
  annote =	{Keywords: Dynamic graphs, set cover, recourse}
}
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.2026.48
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.2025.91
Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism

Authors: Laxman Dhulipala, Monika Henzinger, George Z. Li, Quanquan C. Liu, A. R. Sricharan, and Leqi Zhu

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

Abstract
Many differentially private and classical non-private graph algorithms rely crucially on determining whether some property of each vertex meets a threshold. For example, for the k-core decomposition problem, the classic peeling algorithm iteratively removes a vertex if its induced degree falls below a threshold. The sparse vector technique (SVT) is generally used to transform non-private threshold queries into private ones with only a small additive loss in accuracy. However, a naive application of SVT in the graph setting leads to an amplification of the error by a factor of n due to composition, as SVT is applied to every vertex. In this paper, we resolve this problem by formulating a novel generalized sparse vector technique which we call the Multidimensional AboveThreshold (MAT) Mechanism which generalizes SVT (applied to vectors with one dimension) to vectors with multiple dimensions. When applied to vectors with n dimensions, we solve a number of important graph problems with better bounds than previous work. Specifically, we apply our MAT mechanism to obtain a set of improved bounds for a variety of problems including k-core decomposition, densest subgraph, low out-degree ordering, and vertex coloring. We give a tight local edge differentially private (LEDP) algorithm for k-core decomposition that results in an approximation with O(ε^{-1} log n) additive error and no multiplicative error in O(n) rounds. We also give a new (2+η)-factor multiplicative, O(ε^{-1} log n) additive error algorithm in O(log² n) rounds for any constant η > 0. Both of these results are asymptotically tight against our new lower bound of Ω(log n) for any constant-factor approximation algorithm for k-core decomposition. Our new algorithms for k-core decomposition also directly lead to new algorithms for the related problems of densest subgraph and low out-degree ordering. Finally, we give novel LEDP differentially private defective coloring algorithms that use number of colors given in terms of the arboricity of the graph.
Cite as

Laxman Dhulipala, Monika Henzinger, George Z. Li, Quanquan C. Liu, A. R. Sricharan, and Leqi Zhu. Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 91:1-91:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dhulipala_et_al:LIPIcs.ESA.2025.91,
  author =	{Dhulipala, Laxman and Henzinger, Monika and Li, George Z. and Liu, Quanquan C. and Sricharan, A. R. and Zhu, Leqi},
  title =	{{Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{91:1--91:20},
  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.2025.91},
  URN =		{urn:nbn:de:0030-drops-245601},
  doi =		{10.4230/LIPIcs.ESA.2025.91},
  annote =	{Keywords: differential privacy, abovethreshold, densest subgraph}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.44
Streaming Algorithms for Conflict-Free Coloring

Authors: Rogers Mathew, Fahad Panolan, and Seshikanth

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

Abstract
Conflict-free coloring of a hypergraph ℋ = (V,ℰ) using k colors is a function f:V → {1,2, …, k} such that for all E ∈ ℰ, there exists a vertex v ∈ E with a unique color. That is, f(v)≠ f(u) for all u ∈ E ⧵ {v}. The minimum k for which ℋ has a conflict-free coloring using k colors is called the conflict-free chromatic number of ℋ. For a simple graph G, a conflict-free coloring of the hypergraph with vertex set V(G) and edge set being the set of all closed neighborhoods of the vertices in G is called a conflict-free closed neighborhood (CFCN) coloring of G. CFCN chromatic number, denoted by χ_{CN}(G), is the minimum number of colors used in a conflict-free closed neighborhood coloring of G. Analogously, we define conflict-free open neighborhood (CFON) coloring and CFON chromatic number, χ_{ON}(G), of a graph G. There are various works on proving upper and lower bounds of χ_{ON}(G) and χ_{CN}(G). In this work, we develop streaming algorithms for CFCN and CFON coloring of a graph where the number of colors used matches the best-known upper bounds of χ_{ON}(G) and χi_{CN}(G). Our algorithms use as input an edge stream of the graph G in the insertion-only model. Our results and the best-known bounds for χ_{ON}(G) and χ_{CN}(G) are given below. 1. Pach and Tardos [Combinatorics, Probability and Computing, 2009] showed that, for any n vertex graph G, χ_{CN}(G) = O(ln² n). Glebov, Szabó and Tardos [Combinatorics, Probability and Computing, 2014] showed the existence of graphs G with χ_{CN}(G) = Ω(ln² n). We design a randomized single-pass semi-streaming algorithm (i.e., it uses O(n ln n) space that, given an n-vertex graph G, outputs a CFCN coloring of G using O(ln² n) colors with probability at least (1-2/n). 2. Bhyravarapu, Kalyanasundaram, Mathew [Journal of Graph Theory, 2021] showed that for a graph G with maximum degree Δ, χ_{CN}(G) = O(ln² Δ). The methods used by our algorithms give rise to a simpler, alternate proof for this bound. 3. It is known that χ_{ON}(G) ≤ 1/2 + √{2n + 1/4} (See Pach and Tardos [Combinatorics, Probability and Computing, 2009] and Ph.D. thesis of Cheilaris). This bound is asymptotically tight. - We design a deterministic single-pass O(n√n) space streaming algorithm that, given a graph G on n vertices, finds a CFON coloring using 2√n colors. - We design a randomized, single-pass, semi-streaming algorithm to find a CFON coloring of a graph G using O(√n ln² n) colors with success probability at least (1-2/n). 4. It is known that χ_{ON}(G) ≤ Δ+1, where Δ is the maximum degree of a vertex in G. Further, there are graphs G known with χ_{ON}(G) = Δ + 1. We design a randomized two-pass semi-streaming algorithm (uses O(1/(ε²) n ln³ n) space) that outputs a CFON coloring of G using (1+ε)Δ colors, for any ε > 0, with a probability at least (1-1/n).
Cite as

Rogers Mathew, Fahad Panolan, and Seshikanth. Streaming Algorithms for Conflict-Free Coloring. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mathew_et_al:LIPIcs.WADS.2025.44,
  author =	{Mathew, Rogers and Panolan, Fahad and Seshikanth},
  title =	{{Streaming Algorithms for Conflict-Free Coloring}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{44:1--44:18},
  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.2025.44},
  URN =		{urn:nbn:de:0030-drops-242756},
  doi =		{10.4230/LIPIcs.WADS.2025.44},
  annote =	{Keywords: Streaming algorithm, conflict-free coloring, vertex coloring, randomized algorithms}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.33
A Min-Entropy Approach to Multi-Party Communication Lower Bounds

Authors: Mi-Ying (Miryam) Huang, Xinyu Mao, Shuo Wang, Guangxu Yang, and Jiapeng Zhang

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Information complexity is one of the most powerful techniques to prove information-theoretical lower bounds, in which Shannon entropy plays a central role. Though Shannon entropy has some convenient properties, such as the chain rule, it still has inherent limitations. One of the most notable barriers is the square-root loss, which appears in the square-root gap between entropy gaps and statistical distances, e.g., Pinsker’s inequality. To bypass this barrier, we introduce a new method based on min-entropy analysis. Building on this new method, we prove the following results. - An Ω(N^{∑_i α_i - max_i {α_i}}/k) randomized communication lower bound of the k-party set-intersection problem where the i-th party holds a random set of size ≈ N^{1-α_i}. - A tight Ω(n/k) randomized lower bound of the k-party Tree Pointer Jumping problems, improving an Ω(n/k²) lower bound by Chakrabarti, Cormode, and McGregor (STOC 08). - An Ω(n/k+√n) lower bound of the Chained Index problem, improving an Ω(n/k²) lower bound by Cormode, Dark, and Konrad (ICALP 19). Since these problems served as hard problems for numerous applications in streaming lower bounds and cryptography, our new lower bounds directly improve these streaming lower bounds and cryptography lower bounds. On the technical side, min-entropy does not have nice properties such as the chain rule. To address this issue, we enhance the structure-vs-pseudorandomness decomposition used by Göös, Pitassi, and Watson (FOCS 17) and Yang and Zhang (STOC 24); both papers used this decomposition to prove communication lower bounds. In this paper, we give a new breath to this method in the multi-party setting, presenting a new toolkit for proving multi-party communication lower bounds.
Cite as

Mi-Ying (Miryam) Huang, Xinyu Mao, Shuo Wang, Guangxu Yang, and Jiapeng Zhang. A Min-Entropy Approach to Multi-Party Communication Lower Bounds. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 33:1-33:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{huang_et_al:LIPIcs.CCC.2025.33,
  author =	{Huang, Mi-Ying (Miryam) and Mao, Xinyu and Wang, Shuo and Yang, Guangxu and Zhang, Jiapeng},
  title =	{{A Min-Entropy Approach to Multi-Party Communication Lower Bounds}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{33:1--33:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.33},
  URN =		{urn:nbn:de:0030-drops-237273},
  doi =		{10.4230/LIPIcs.CCC.2025.33},
  annote =	{Keywords: communication complexity, lifting theorems, set intersection, chained index}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.16
Direct Sums for Parity Decision Trees

Authors: Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Direct sum theorems state that the cost of solving k instances of a problem is at least Ω(k) times the cost of solving a single instance. We prove the first such results in the randomised parity decision tree model. We show that a direct sum theorem holds whenever (1) the lower bound for parity decision trees is proved using the discrepancy method; or (2) the lower bound is proved relative to a product distribution.
Cite as

Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan. Direct Sums for Parity Decision Trees. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{besselman_et_al:LIPIcs.CCC.2025.16,
  author =	{Besselman, Tyler and G\"{o}\"{o}s, Mika and Guo, Siyao and Maystre, Gilbert and Yuan, Weiqiang},
  title =	{{Direct Sums for Parity Decision Trees}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{16:1--16:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.16},
  URN =		{urn:nbn:de:0030-drops-237105},
  doi =		{10.4230/LIPIcs.CCC.2025.16},
  annote =	{Keywords: direct sum, parity decision trees, query complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.103
Approximating Dasgupta Cost in Sublinear Time from a Few Random Seeds

Authors: Michael Kapralov, Akash Kumar, Silvio Lattanzi, Aida Mousavifar, and Weronika Wrzos-Kaminska

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

Abstract
Testing graph cluster structure has been a central object of study in property testing since the foundational work of Goldreich and Ron [STOC'96] on expansion testing, i.e. the problem of distinguishing between a single cluster (an expander) and a graph that is far from a single cluster. More generally, a (k, ε)-clusterable graph G is a graph whose vertex set admits a partition into k induced expanders, each with outer conductance bounded by ε. A recent line of work initiated by Czumaj, Peng and Sohler [STOC'15] has shown how to test whether a graph is close to (k, ε)-clusterable, and to locally determine which cluster a given vertex belongs to with misclassification rate ≈ ε, but no sublinear time algorithms for learning the structure of inter-cluster connections are known. As a simple example, can one locally distinguish between the "cluster graph" forming a line and a clique? In this paper, we consider the problem of testing the hierarchical cluster structure of (k, ε)-clusterable graphs in sublinear time. Our measure of hierarchical clusterability is the well-established Dasgupta cost, and our main result is an algorithm that approximates Dasgupta cost of a (k, ε)-clusterable graph in sublinear time, using a small number of randomly chosen seed vertices for which cluster labels are known. Our main result is an O(√{log k}) approximation to Dasgupta cost of G in ≈ n^{1/2+O(ε)} time using ≈ n^{1/3} seeds, effectively giving a sublinear time simulation of the algorithm of Charikar and Chatziafratis [SODA'17] on clusterable graphs. To the best of our knowledge, ours is the first result on approximating the hierarchical clustering properties of such graphs in sublinear time.
Cite as

Michael Kapralov, Akash Kumar, Silvio Lattanzi, Aida Mousavifar, and Weronika Wrzos-Kaminska. Approximating Dasgupta Cost in Sublinear Time from a Few Random Seeds. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 103:1-103:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kapralov_et_al:LIPIcs.ICALP.2025.103,
  author =	{Kapralov, Michael and Kumar, Akash and Lattanzi, Silvio and Mousavifar, Aida and Wrzos-Kaminska, Weronika},
  title =	{{Approximating Dasgupta Cost in Sublinear Time from a Few Random Seeds}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{103:1--103:19},
  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.2025.103},
  URN =		{urn:nbn:de:0030-drops-234804},
  doi =		{10.4230/LIPIcs.ICALP.2025.103},
  annote =	{Keywords: Sublinear algorithms, Hierarchical Clustering, Dasgupta’s Cost}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.119
Faster Semi-Streaming Matchings via Alternating Trees

Authors: Slobodan Mitrović, Anish Mukherjee, Piotr Sankowski, and Wen-Horng Sheu

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

Abstract
We design a deterministic algorithm for the (1+ε)-approximate maximum matching problem. Our primary result demonstrates that this problem can be solved in O(ε^{-6}) semi-streaming passes, improving upon the O(ε^{-19}) pass-complexity algorithm by [Fischer, Mitrović, and Uitto, STOC'22]. This contributes substantially toward resolving Open question 2 from [Assadi, SOSA'24]. Leveraging the framework introduced in [FMU'22], our algorithm achieves an analogous round complexity speed-up for computing a (1+ε)-approximate maximum matching in both the Massively Parallel Computation (MPC) and CONGEST models. The data structures maintained by our algorithm are formulated using blossom notation and represented through alternating trees. This approach enables a simplified correctness analysis by treating specific components as if operating on bipartite graphs, effectively circumventing certain technical intricacies present in prior work.
Cite as

Slobodan Mitrović, Anish Mukherjee, Piotr Sankowski, and Wen-Horng Sheu. Faster Semi-Streaming Matchings via Alternating Trees. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 119:1-119:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mitrovic_et_al:LIPIcs.ICALP.2025.119,
  author =	{Mitrovi\'{c}, Slobodan and Mukherjee, Anish and Sankowski, Piotr and Sheu, Wen-Horng},
  title =	{{Faster Semi-Streaming Matchings via Alternating Trees}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{119:1--119:19},
  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.2025.119},
  URN =		{urn:nbn:de:0030-drops-234965},
  doi =		{10.4230/LIPIcs.ICALP.2025.119},
  annote =	{Keywords: streaming algorithms, approximation algorithms, maximum matching}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.100
Robust-Sorting and Applications to Ulam-Median

Authors: Ragesh Jaiswal, Amit Kumar, and Jatin Yadav

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

Abstract
Sorting is one of the most basic primitives in many algorithms and data analysis tasks. Comparison-based sorting algorithms, like quick-sort and merge-sort, are known to be optimal when the outcome of each comparison is error-free. However, many real-world sorting applications operate in scenarios where the outcome of each comparison can be noisy. In this work, we explore settings where a bounded number of comparisons are potentially corrupted by erroneous agents, resulting in arbitrary, adversarial outcomes. We model the sorting problem as a query-limited tournament graph where edges involving erroneous nodes may yield arbitrary results. Our primary contribution is a randomized algorithm inspired by quick-sort that, in expectation, produces an ordering close to the true total order while only querying Õ(n) edges. We achieve a distance from the target order π within (3 + ε)|B|, where B is the set of erroneous nodes, balancing the competing objectives of minimizing both query complexity and misalignment with π. Our algorithm needs to carefully balance two aspects - identify a pivot that partitions the vertex set evenly and ensure that this partition is "truthful" and yet query as few "triangles" in the graph G as possible. Since the nodes in B can potentially hide in an intricate manner, our algorithm requires several technical steps that ensure that progress is made in each recursive step. Additionally, we demonstrate significant implications for the Ulam-k-Median problem. This is a classical clustering problem where the metric is defined on the set of permutations on a set of d elements. Chakraborty, Das, and Krauthgamer gave a (2-ε) FPT approximation algorithm for this problem, where the running time is super-linear in both n and d. We give the first (2-ε) FPT linear time approximation algorithm for this problem. Our main technical result gives a strengthening of the results in Chakraborty et al. by showing that a good 1-median solution can be obtained from a constant-size random sample of the input. We use our robust sorting framework to find a good solution from such a random sample. We feel that the notion of robust sorting should have applications in several such settings.
Cite as

Ragesh Jaiswal, Amit Kumar, and Jatin Yadav. Robust-Sorting and Applications to Ulam-Median. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jaiswal_et_al:LIPIcs.ICALP.2025.100,
  author =	{Jaiswal, Ragesh and Kumar, Amit and Yadav, Jatin},
  title =	{{Robust-Sorting and Applications to Ulam-Median}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{100:1--100:19},
  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.2025.100},
  URN =		{urn:nbn:de:0030-drops-234774},
  doi =		{10.4230/LIPIcs.ICALP.2025.100},
  annote =	{Keywords: Sorting, clustering, query complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.10
Faster Counting and Sampling Algorithms Using Colorful Decision Oracle

Authors: Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
In this work, we consider d-Hyperedge Estimation and d-Hyperedge Sample problem in a hypergraph H(U(H),F(H)) in the query complexity framework, where U(H) denotes the set of vertices and F(H) denotes the set of hyperedges. The oracle access to the hypergraph is called Colorful Independence Oracle (CID), which takes d (non-empty) pairwise disjoint subsets of vertices A₁,…, A_d ⊆ U(ℋ) as input, and answers whether there exists a hyperedge in H having (exactly) one vertex in each A_i, i ∈ {1,2,…,d}. The problem of d-Hyperedge Estimation and d-Hyperedge Sample with CID oracle access is important in its own right as a combinatorial problem. Also, Dell et al. [SODA '20] established that decision vs counting complexities of a number of combinatorial optimization problems can be abstracted out as d-Hyperedge Estimation problems with a CID oracle access. The main technical contribution of the paper is an algorithm that estimates m = |F(H)| with m̂ such that 1/(C_{d)log^{d-1} n) ≤ m̂/m ≤ C_{d} log ^{d-1} n. by using at most C_{d}log ^{d+2} n many CID queries, where n denotes the number of vertices in the hypergraph H and C_d is a constant that depends only on d}. Our result coupled with the framework of Dell et al. [SODA '21] implies improved bounds for the following fundamental problems: Edge Estimation using the Bipartite Independent Set (BIS). We improve the bound obtained by Beame et al. [ITCS '18, TALG '20]. Triangle Estimation using the Tripartite Independent Set (TIS). The previous best bound for the case of graphs with low co-degree (Co-degree for an edge in the graph is the number of triangles incident to that edge in the graph) was due to Bhattacharya et al. [ISAAC '19, TOCS '21], and Dell {et al.}’s result gives the best bound for the case of general graphs [SODA '21]. We improve both of these bounds. Hyperedge Estimation & Sampling using Colorful Independence Oracle (CID). We give an improvement over the bounds obtained by Dell et al. [SODA '21].
Cite as

Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Faster Counting and Sampling Algorithms Using Colorful Decision Oracle. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhattacharya_et_al:LIPIcs.STACS.2022.10,
  author =	{Bhattacharya, Anup and Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath},
  title =	{{Faster Counting and Sampling Algorithms Using Colorful Decision Oracle}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.10},
  URN =		{urn:nbn:de:0030-drops-158205},
  doi =		{10.4230/LIPIcs.STACS.2022.10},
  annote =	{Keywords: Query Complexity, Subset Query, Hyperedge Estimation, and Colorful Independent Set oracle}
}
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.ITCS.2021.15
Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!

Authors: Anup Bhattacharya, Arijit Bishnu, Gopinath Mishra, and Anannya Upasana

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We study a graph coloring problem that is otherwise easy in the RAM model but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our main work is on estimating the number of conflicting or monochromatic edges given a coloring function that is streaming along with the graph; we call the problem Conflict-Est. The coloring function on a vertex can be read or accessed only when the vertex is revealed in the stream. If we need the color on a vertex that has streamed past, then that color, along with its vertex, has to be stored explicitly. We provide algorithms for a graph that is streaming in different variants of the vertex arrival in one-pass streaming model, viz. the Vertex Arrival (VA), Vertex Arrival With Degree Oracle (VAdeg), Vertex Arrival in Random Order (VArand) models, with special focus on the random order model. We also provide matching lower bounds for most of the cases. The mainstay of our work is in showing that the properties of a random order stream can be exploited to design efficient streaming algorithms for estimating the number of monochromatic edges. We have also obtained a lower bound, though not matching the upper bound, for the random order model. Among all the three models vis-a-vis this problem, we can show a clear separation of power in favor of the VArand model.
Cite as

Anup Bhattacharya, Arijit Bishnu, Gopinath Mishra, and Anannya Upasana. Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhattacharya_et_al:LIPIcs.ITCS.2021.15,
  author =	{Bhattacharya, Anup and Bishnu, Arijit and Mishra, Gopinath and Upasana, Anannya},
  title =	{{Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.15},
  URN =		{urn:nbn:de:0030-drops-135544},
  doi =		{10.4230/LIPIcs.ITCS.2021.15},
  annote =	{Keywords: Streaming, random ordering, graph coloring, estimation, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.14
FPT Approximation for Constrained Metric k-Median/Means

Authors: Dishant Goyal, Ragesh Jaiswal, and Amit Kumar

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
The Metric k-median problem over a metric space (𝒳, d) is defined as follows: given a set L ⊆ 𝒳 of facility locations and a set C ⊆ 𝒳 of clients, open a set F ⊆ L of k facilities such that the total service cost, defined as Φ(F, C) := ∑_{x ∈ C} min_{f ∈ F} d(x, f), is minimised. The metric k-means problem is defined similarly using squared distances (i.e., d²(., .) instead of d(., .)). In many applications there are additional constraints that any solution needs to satisfy. For example, to balance the load among the facilities in resource allocation problems, a capacity u is imposed on every facility. That is, no more than u clients can be assigned to any facility. This problem is known as the capacitated k-means/k-median problem. Likewise, various other applications have different constraints, which give rise to different constrained versions of the problem such as r-gather, fault-tolerant, outlier k-means/k-median problem. Surprisingly, for many of these constrained problems, no constant-approximation algorithm is known. Moreover, the unconstrained problem itself is known [Marek Adamczyk et al., 2019] to be W[2]-hard when parameterized by k. We give FPT algorithms with constant approximation guarantee for a range of constrained k-median/means problems. For some of the constrained problems, ours is the first constant factor approximation algorithm whereas for others, we improve or match the approximation guarantee of previous works. We work within the unified framework of Ding and Xu [Ding and Xu, 2015] that allows us to simultaneously obtain algorithms for a range of constrained problems. In particular, we obtain a (3+ε)-approximation and (9+ε)-approximation for the constrained versions of the k-median and k-means problem respectively in FPT time. In many practical settings of the k-median/means problem, one is allowed to open a facility at any client location, i.e., C ⊆ L. For this special case, our algorithm gives a (2+ε)-approximation and (4+ε)-approximation for the constrained versions of k-median and k-means problem respectively in FPT time. Since our algorithm is based on simple sampling technique, it can also be converted to a constant-pass log-space streaming algorithm. In particular, here are some of the main highlights of this work: 1) For the uniform capacitated k-median/means problems our results matches previously known results of Addad et al. [Vincent Cohen-Addad and Jason Li, 2019]. 2) For the r-gather k-median/means problem (clustering with lower bound on the size of clusters), our FPT approximation bounds are better than what was previously known. 3) Our approximation bounds for the fault-tolerant, outlier, and uncertain versions is better than all previously known results, albeit in FPT time. 4) For certain constrained settings such as chromatic, l-diversity, and semi-supervised k-median/means, we obtain the first constant factor approximation algorithms to the best of our knowledge. 5) Since our algorithms are based on a simple sampling based approach, we also obtain constant-pass log-space streaming algorithms for most of the above-mentioned problems.
Cite as

Dishant Goyal, Ragesh Jaiswal, and Amit Kumar. FPT Approximation for Constrained Metric k-Median/Means. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{goyal_et_al:LIPIcs.IPEC.2020.14,
  author =	{Goyal, Dishant and Jaiswal, Ragesh and Kumar, Amit},
  title =	{{FPT Approximation for Constrained Metric k-Median/Means}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.14},
  URN =		{urn:nbn:de:0030-drops-133170},
  doi =		{10.4230/LIPIcs.IPEC.2020.14},
  annote =	{Keywords: k-means, k-median, approximation algorithms, parameterised algorithms}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.13
On Sampling Based Algorithms for k-Means

Authors: Anup Bhattacharya, Dishant Goyal, Ragesh Jaiswal, and Amit Kumar

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We generalise the results of Bhattacharya et al. [Bhattacharya et al., 2018] for the list-k-means problem defined as - for a (unknown) partition X₁, ..., X_k of the dataset X ⊆ ℝ^d, find a list of k-center-sets (each element in the list is a set of k centers) such that at least one of k-center-sets {c₁, ..., c_k} in the list gives an (1+ε)-approximation with respect to the cost function min_{permutation π} [∑_{i = 1}^{k} ∑_{x ∈ X_i} ||x - c_{π(i)}||²]. The list-k-means problem is important for the constrained k-means problem since algorithms for the former can be converted to {PTAS} for various versions of the latter. The algorithm for the list-k-means problem by Bhattacharya et al. is a D²-sampling based algorithm that runs in k iterations. Making use of a constant factor solution for the (classical or unconstrained) k-means problem, we generalise the algorithm of Bhattacharya et al. in two ways - (i) for any fixed set X_{j₁}, ..., X_{j_t} of t ≤ k clusters, the algorithm produces a list of (k/(ε))^{O(t/(ε))} t-center sets such that (w.h.p.) at least one of them is good for X_{j₁}, ..., X_{j_t}, and (ii) the algorithm runs in a single iteration. Following are the consequences of our generalisations: 1) Faster PTAS under stability and a parameterised reduction: Property (i) of our generalisation is useful in scenarios where finding good centers becomes easier once good centers for a few "bad" clusters have been chosen. One such case is clustering under stability of Awasthi et al. [Awasthi et al., 2010] where the number of such bad clusters is a constant. Using property (i), we significantly improve the running time of their algorithm from O(dn³) (k log{n})^{poly(1/(β), 1/(ε))} to O (dn³ (k/(ε)) ^{O(1/βε²)}). Another application is a parameterised reduction from the outlier version of k-means to the classical one where the bad clusters are the outliers. 2) Streaming algorithms: The sampling algorithm running in a single iteration (i.e., property (ii)) allows us to design a constant-pass, logspace streaming algorithm for the list-k-means problem. This can be converted to a constant-pass, logspace streaming PTAS for various constrained versions of the k-means problem. In particular, this gives a 3-pass, polylog-space streaming PTAS for the constrained binary k-means problem which in turn gives a 4-pass, polylog-space streaming PTAS for the generalised binary 𝓁₀-rank-r approximation problem. This is the first constant pass, polylog-space streaming algorithm for either of the two problems. Coreset based techniques, which is another approach for designing streaming algorithms in general, is not known to work for the constrained binary k-means problem to the best of our knowledge.
Cite as

Anup Bhattacharya, Dishant Goyal, Ragesh Jaiswal, and Amit Kumar. On Sampling Based Algorithms for k-Means. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bhattacharya_et_al:LIPIcs.FSTTCS.2020.13,
  author =	{Bhattacharya, Anup and Goyal, Dishant and Jaiswal, Ragesh and Kumar, Amit},
  title =	{{On Sampling Based Algorithms for k-Means}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.13},
  URN =		{urn:nbn:de:0030-drops-132549},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.13},
  annote =	{Keywords: k-means, low rank approximation}
}
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