32 Search Results for "Schwiegelshohn, Chris"

Document
DOI: 10.4230/LIPIcs.SWAT.2026.19
Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means

Authors: Nicole Funk, Annika Hennes, Johanna Hillebrand, and Sarah Sturm

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
We study discrete k-clustering problems in general metric spaces that are constrained by a combination of two different fairness conditions within the demographic fairness model. Given a metric space (P,d), where every point in P is equipped with a protected attribute, and a number k, the goal is to partition P into k clusters with a designated center each, such that a center-based objective function is minimized and the attributes are fairly distributed with respect to the following two fairness concepts: 1) group fairness: We aim for clusters with balanced numbers of attributes by specifying lower and upper bounds for the desired attribute proportions. 2) diverse center selection: Clusters have natural representatives, i.e., their centers. We ask for a balanced set of representatives by specifying the desired number of centers to choose from each attribute. Dickerson, Esmaeili, Morgenstern, and Pena [John P. Dickerson et al., 2023] denote the combination of these two constraints as doubly constrained fair clustering. They present algorithms whose guarantees depend on the best known approximation factors for either of these problems. Currently, this implies an 8-approximation with a small additive violation on the group fairness constraint. For k-center, we improve this approximation factor to 4 with a small additive violation. This guarantee also depends on the currently best algorithm for DS-fair k-center given by Jones, Nguyen and Nguyen [Matthew Jones et al., 2020]. For k-median and k-means, we propose the first constant-factor approximation algorithms. Our algorithms transform a solution that satisfies diverse center selection into a doubly constrained fair clustering using an LP-based approach. Furthermore, our results are generalizable to other center-selection constraints, such as matroid k-clustering and knapsack constraints.
Cite as

Nicole Funk, Annika Hennes, Johanna Hillebrand, and Sarah Sturm. Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{funk_et_al:LIPIcs.SWAT.2026.19,
  author =	{Funk, Nicole and Hennes, Annika and Hillebrand, Johanna and Sturm, Sarah},
  title =	{{Constant-Factor Approximations for Doubly Constrained Fair k-Center, k-Median and k-Means}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.19},
  URN =		{urn:nbn:de:0030-drops-260551},
  doi =		{10.4230/LIPIcs.SWAT.2026.19},
  annote =	{Keywords: Clustering, Fairness, Approximation Algorithms, k-center, k-median, k-means}
}
Document
DOI: 10.4230/LIPIcs.FORC.2026.10
Packing Compact Subgraphs with Applications to Districting

Authors: Ho-Lin Chen, Po-Yu Chou, Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract
Packing disjoint subgraphs in a given graph is a fundamental problem with many applications. Motivated by political districting, we focus on connected subgraphs that are compact (e.g., having constant radius from a single center vertex) and that satisfy additional composition requirements, such as a minimum population/weight threshold or balanced weight types (e.g., political affiliations). We aim to maximize coverage by disjoint districts that meet these requirements. In this work, we present new results that substantially improve the previously known bounds on balanced star districts for planar and minor-free graphs [Prathamesh Dharangutte et al., 2025]. In particular, we improve the approximation factor from O(log n) to O(1) for packing balanced star districts using the exact same algorithm, but with a refined analysis. We also extend the results beyond planar graphs to minor-free graphs and an even broader family of graphs of bounded expansion. Additionally, we obtain an O(1) approximation for packing radius-k districts (with a constant k) in planar and apex-minor-free graphs. In order to get a (1+ε) approximation on the max coverage, we show that this can be achieved if we allow a slight relaxation of the balancedness parameters (by a factor that can be made arbitrarily close to 1), for bounded radius-k districts on planar and apex-minor-free graphs. We show that all of these results can also be obtained if we enforce a minimum weight threshold for each district as the composition requirement, rather than balancedness. We present various results on hardness and hardness of approximation for this variant, by graph and district types.
Cite as

Ho-Lin Chen, Po-Yu Chou, Prathamesh Dharangutte, Jie Gao, Shang-En Huang, and Fang-Yi Yu. Packing Compact Subgraphs with Applications to Districting. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 10:1-10:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chen_et_al:LIPIcs.FORC.2026.10,
  author =	{Chen, Ho-Lin and Chou, Po-Yu and Dharangutte, Prathamesh and Gao, Jie and Huang, Shang-En and Yu, Fang-Yi},
  title =	{{Packing Compact Subgraphs with Applications to Districting}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{10:1--10:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.10},
  URN =		{urn:nbn:de:0030-drops-259820},
  doi =		{10.4230/LIPIcs.FORC.2026.10},
  annote =	{Keywords: Approximation algorithms, algorithmic fairness}
}
Document
DOI: 10.4230/LIPIcs.FORC.2026.1
Computational Hardness of Private Coreset

Authors: Badih Ghazi, Cristóbal Guzmán, Pritish Kamath, Alexander Knop, Ravi Kumar, and Pasin Manurangsi

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract
We study the problem of differentially private (DP) computation of coreset for the k-means objective. For a given input set of points, a coreset is another set of points such that the k-means objective for any candidate solution is preserved up to a multiplicative (1 ± α) factor (and some additive factor). We prove the first computational lower bounds for this problem. Specifically, assuming the existence of one-way functions, we show that no polynomial-time (ε, 1/n^{ω(1)})-DP algorithm can compute a coreset for k-means in the 𝓁_∞-metric for some constant α > 0 (and some constant additive factor), even for k = 3. For k-means in the Euclidean metric, we show a similar result but only for α = Θ(1/d²), where d is the dimension.
Cite as

Badih Ghazi, Cristóbal Guzmán, Pritish Kamath, Alexander Knop, Ravi Kumar, and Pasin Manurangsi. Computational Hardness of Private Coreset. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghazi_et_al:LIPIcs.FORC.2026.1,
  author =	{Ghazi, Badih and Guzm\'{a}n, Crist\'{o}bal and Kamath, Pritish and Knop, Alexander and Kumar, Ravi and Manurangsi, Pasin},
  title =	{{Computational Hardness of Private Coreset}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{1:1--1:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.1},
  URN =		{urn:nbn:de:0030-drops-259725},
  doi =		{10.4230/LIPIcs.FORC.2026.1},
  annote =	{Keywords: Differentially Private Clustering, Coreset, Cryptographic Hardness}
}
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.2026.80
Hardness of High-Dimensional Linear Classification

Authors: Alexander Munteanu, Simon Omlor, and Jeff M. Phillips

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

Abstract
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only O(n^d) and respectively Õ(1/ε^d) upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and k-Sum problems. Our reductions yield matching lower bounds of Ω̃(n^d) and respectively Ω̃(1/ε^d) based on Affine Degeneracy testing, and Ω̃(n^{d/2}) and respectively Ω̃(1/ε^{d/2}) conditioned on k-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.
Cite as

Alexander Munteanu, Simon Omlor, and Jeff M. Phillips. Hardness of High-Dimensional Linear Classification. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{munteanu_et_al:LIPIcs.SoCG.2026.80,
  author =	{Munteanu, Alexander and Omlor, Simon and Phillips, Jeff M.},
  title =	{{Hardness of High-Dimensional Linear Classification}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{80:1--80:16},
  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.2026.80},
  URN =		{urn:nbn:de:0030-drops-258871},
  doi =		{10.4230/LIPIcs.SoCG.2026.80},
  annote =	{Keywords: Conditional Hardness, k-Sum, Affine Degeneracy, Halfspace Discrepancy, Classification}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.82
Dimension Reduction for Clustering: The Curious Case of Discrete Centers

Authors: Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of n points into dimension O(log n) with low distortion of their distances. This dimension bound is tight in general, but one can bypass it for specific problems. Indeed, tremendous progress has been made for clustering problems, especially in the continuous setting where centers can be picked from the ambient space ℝ^d. Most notably, for k-median and k-means, the dimension bound was improved to O(log k) [Makarychev, Makarychev and Razenshteyn, STOC 2019]. We explore dimension reduction for clustering in the discrete setting, where centers can only be picked from the dataset, and present two results that are both parameterized by the doubling dimension of the dataset, denoted as ddim. The first result shows that dimension O_{ε}(ddim + log k + log log n) suffices, and is moreover tight, to guarantee that the cost is preserved within factor 1±ε for every set of centers. Our second result eliminates the log log n term in the dimension through a relaxation of the guarantee (namely, preserving the cost only for all approximately-optimal sets of centers), which maintains its usefulness for downstream applications. Overall, we achieve strong dimension reduction in the discrete setting, and find that it differs from the continuous setting not only in the dimension bound, which depends on the doubling dimension, but also in the guarantees beyond preserving the optimal value, such as which clusterings are preserved.
Cite as

Shaofeng H.-C. Jiang, Robert Krauthgamer, Shay Sapir, Sandeep Silwal, and Di Yue. Dimension Reduction for Clustering: The Curious Case of Discrete Centers. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 82:1-82:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jiang_et_al:LIPIcs.ITCS.2026.82,
  author =	{Jiang, Shaofeng H.-C. and Krauthgamer, Robert and Sapir, Shay and Silwal, Sandeep and Yue, Di},
  title =	{{Dimension Reduction for Clustering: The Curious Case of Discrete Centers}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{82:1--82:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.82},
  URN =		{urn:nbn:de:0030-drops-253698},
  doi =		{10.4230/LIPIcs.ITCS.2026.82},
  annote =	{Keywords: dimension reduction, clustering, k-median, k-means, doubling dimension}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2025.8
Binary k-Center with Missing Entries: Structure Leads to Tractability

Authors: Tobias Friedrich, Kirill Simonov, and Farehe Soheil

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

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

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

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@InProceedings{friedrich_et_al:LIPIcs.IPEC.2025.8,
  author =	{Friedrich, Tobias and Simonov, Kirill and Soheil, Farehe},
  title =	{{Binary k-Center with Missing Entries: Structure Leads to Tractability}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.8},
  URN =		{urn:nbn:de:0030-drops-251403},
  doi =		{10.4230/LIPIcs.IPEC.2025.8},
  annote =	{Keywords: Clustering, Missing Entries, k-Center, Parameterized Algorithms}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.19
Clustering in Varying Metrics

Authors: Deeparnab Chakrabarty, Jonathan Conroy, and Ankita Sarkar

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
We introduce the aggregated clustering problem, where one is given T instances of a center-based clustering task over the same n points, but under different metrics. The goal is to open k centers to minimize an aggregate of the clustering costs - e.g., the average or maximum - where the cost is measured via k-center/median/means objectives. More generally, we minimize a norm Ψ over the T cost values. We show that for T ≥ 3, the problem is inapproximable to any finite factor in polynomial time. For T = 2, we give constant-factor approximations. We also show W[2]-hardness when parameterized by k, but obtain f(k,T)poly(n)-time 3-approximations when parameterized by both k and T. When the metrics have structure, we obtain efficient parameterized approximation schemes (EPAS). If all T metrics have bounded ε-scatter dimension, we achieve a (1+ε)-approximation in f(k,T,ε)poly(n) time. If the metrics are induced by edge weights on a common graph G of bounded treewidth tw, and Ψ is the sum function, we get an EPAS in f(T,ε,tw)poly(n,k) time. Conversely, unless (randomized) ETH is false, any finite factor approximation is impossible if parametrized by only T, even when the treewidth is tw = Ω(polylog n).
Cite as

Deeparnab Chakrabarty, Jonathan Conroy, and Ankita Sarkar. Clustering in Varying Metrics. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chakrabarty_et_al:LIPIcs.FSTTCS.2025.19,
  author =	{Chakrabarty, Deeparnab and Conroy, Jonathan and Sarkar, Ankita},
  title =	{{Clustering in Varying Metrics}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{19:1--19:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.19},
  URN =		{urn:nbn:de:0030-drops-251007},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.19},
  annote =	{Keywords: Clustering, approximation algorithms, LP rounding, parameterized and exact algorithms, dynamic programming, fixed parameter tractability, hardness of approximation}
}
Document
DOI: 10.4230/LIPIcs.DISC.2025.40
On the Randomized Locality of Matching Problems in Regular Graphs

Authors: Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract
The main goal in distributed symmetry-breaking is to understand the locality of problems: the radius of the neighborhood that a node must explore to determine its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. Our main results are summarized as follows: 1) Approximate matching: We develop randomized algorithms to show that (1 + ε)-approximate matching in regular graphs is truly local, i.e., the locality depends only on ε and is independent of all other graph parameters. Furthermore, as long as the degree Δ is not very small (namely, as long as Δ ≥ poly(1/ε)), this dependence is only logarithmic in 1/ε. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes n or the degree Δ. 2) Maximal matching: Our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only O(1). Central to our main technical contribution is a novel martingale-based analysis for the ≈ 40-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby’s algorithm on the line graph of a Δ-regular graph results in an almost Δ/2-regular graph.
Cite as

Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang. On the Randomized Locality of Matching Problems in Regular Graphs. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{khoury_et_al:LIPIcs.DISC.2025.40,
  author =	{Khoury, Seri and Purohit, Manish and Schild, Aaron and Wang, Joshua R.},
  title =	{{On the Randomized Locality of Matching Problems in Regular Graphs}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.40},
  URN =		{urn:nbn:de:0030-drops-248570},
  doi =		{10.4230/LIPIcs.DISC.2025.40},
  annote =	{Keywords: regular graphs, maximum matching, augmenting paths, distributed algorithms, Luby’s algorithm, martingales}
}
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
DOI: 10.4230/LIPIcs.ESA.2025.63
Connected k-Median with Disjoint and Non-Disjoint Clusters

Authors: Jan Eube, Kelin Luo, Dorian Reineccius, Heiko Röglin, and Melanie Schmidt

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

Abstract
The connected k-median problem is a constrained clustering problem that combines distance-based k-clustering with connectivity information. The problem allows to input a metric space and an unweighted undirected connectivity graph that is completely unrelated to the metric space. The goal is to compute k centers and corresponding clusters such that each cluster forms a connected subgraph of G, and such that the k-median cost is minimized. The problem has applications in very different fields like geodesy (particularly districting), social network analysis (especially community detection), or bioinformatics. We study a version with overlapping clusters where points can be part of multiple clusters which is natural for the use case of community detection. This problem variant is Ω(log n)-hard to approximate, and our main result is an 𝒪(k² log n)-approximation algorithm for the problem. We complement it with an Ω(n^{1-ε})-hardness result for the case of disjoint clusters without overlap with general connectivity graphs, as well as an exact algorithm in this setting if the connectivity graph is a tree.
Cite as

Jan Eube, Kelin Luo, Dorian Reineccius, Heiko Röglin, and Melanie Schmidt. Connected k-Median with Disjoint and Non-Disjoint Clusters. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eube_et_al:LIPIcs.ESA.2025.63,
  author =	{Eube, Jan and Luo, Kelin and Reineccius, Dorian and R\"{o}glin, Heiko and Schmidt, Melanie},
  title =	{{Connected k-Median with Disjoint and Non-Disjoint Clusters}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{63:1--63:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.63},
  URN =		{urn:nbn:de:0030-drops-245317},
  doi =		{10.4230/LIPIcs.ESA.2025.63},
  annote =	{Keywords: Clustering, Connectivity constraints, Approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.39
Going Beyond Surfaces in Diameter Approximation

Authors: Michał Włodarczyk

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

Abstract
Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge weights, there are known (1+ε)-approximation algorithms with running time poly(1/ε, log n)⋅ n. However, these algorithms rely on shortest path separators and this technique falls short to yield efficient algorithms beyond graphs of bounded genus. In this work we depart from embedding-based arguments and obtain diameter approximations relying on VC set systems and the local treewidth property. We present two orthogonal extensions of the planar case by giving (1+ε)-approximation algorithms with the following running times: - 𝒪_h((1/ε)^𝒪(h) ⋅ nlog² n)-time algorithm for graphs excluding an apex graph of size h as a minor, - 𝒪_d((1/ε)^𝒪(d) ⋅ nlog² n)-time algorithm for the class of d-apex graphs. As a stepping stone, we obtain efficient (1+ε)-approximate distance oracles for graphs excluding an apex graph of size h as a minor. Our oracle has preprocessing time 𝒪_h((1/ε)⁸⋅ nlog nlog W) and query time 𝒪_h((1/ε)²⋅log n log W), where W is the metric stretch. Such oracles have been so far only known for bounded genus graphs. All our algorithms are deterministic.
Cite as

Michał Włodarczyk. Going Beyond Surfaces in Diameter Approximation. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wlodarczyk:LIPIcs.ESA.2025.39,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Going Beyond Surfaces in Diameter Approximation}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{39:1--39:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.39},
  URN =		{urn:nbn:de:0030-drops-245076},
  doi =		{10.4230/LIPIcs.ESA.2025.39},
  annote =	{Keywords: diameter, approximation, distance oracles, graph minors, treewidth}
}
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
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.2
Covering Simple Orthogonal Polygons with Rectangles

Authors: Aniket Basu Roy

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

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

Aniket Basu Roy. Covering Simple Orthogonal Polygons with Rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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