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Documents authored by Conroy, Jonathan

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).
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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.SoCG.2024.37
Optimal Euclidean Tree Covers

Authors: Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-2d}). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set. In this work we present a (1+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a (1+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.
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Hsien-Chih Chang, Jonathan Conroy, Hung Le, Lazar Milenković, Shay Solomon, and Cuong Than. Optimal Euclidean Tree Covers. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chang_et_al:LIPIcs.SoCG.2024.37,
  author =	{Chang, Hsien-Chih and Conroy, Jonathan and Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Than, Cuong},
  title =	{{Optimal Euclidean Tree Covers}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.37},
  URN =		{urn:nbn:de:0030-drops-199828},
  doi =		{10.4230/LIPIcs.SoCG.2024.37},
  annote =	{Keywords: Tree cover, spanner, Steiner point, routing, bounded-degree, quadtree, net-tree}
}
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