# Search Results

### Documents authored by Klimenko, Georgiy

Document
##### Clustering with Neighborhoods

Authors: Hongyao Huang, Georgiy Klimenko, and Benjamin Raichel

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

##### Abstract
In the standard planar k-center clustering problem, one is given a set P of n points in the plane, and the goal is to select k center points, so as to minimize the maximum distance over points in P to their nearest center. Here we initiate the systematic study of the clustering with neighborhoods problem, which generalizes the k-center problem to allow the covered objects to be a set of general disjoint convex objects C rather than just a point set P. For this problem we first show that there is a PTAS for approximating the number of centers. Specifically, if r_opt is the optimal radius for k centers, then in n^O(1/ε²) time we can produce a set of (1+ε)k centers with radius ≤ r_opt. If instead one considers the standard goal of approximating the optimal clustering radius, while keeping k as a hard constraint, we show that the radius cannot be approximated within any factor in polynomial time unless P = NP, even when C is a set of line segments. When C is a set of unit disks we show the problem is hard to approximate within a factor of (√{13}-√3)(2-√3) ≈ 6.99. This hardness result complements our main result, where we show that when the objects are disks, of possibly differing radii, there is a (5+2√3)≈ 8.46 approximation algorithm. Additionally, for unit disks we give an O(n log k)+(k/ε)^O(k) time (1+ε)-approximation to the optimal radius, that is, an FPTAS for constant k whose running time depends only linearly on n. Finally, we show that the one dimensional version of the problem, even when intersections are allowed, can be solved exactly in O(n log n) time.

##### Cite as

Hongyao Huang, Georgiy Klimenko, and Benjamin Raichel. Clustering with Neighborhoods. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

```@InProceedings{huang_et_al:LIPIcs.ISAAC.2021.6,
author =	{Huang, Hongyao and Klimenko, Georgiy and Raichel, Benjamin},
title =	{{Clustering with Neighborhoods}},
booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
pages =	{6:1--6:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-214-3},
ISSN =	{1868-8969},
year =	{2021},
volume =	{212},
editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.6},
URN =		{urn:nbn:de:0030-drops-154398},
doi =		{10.4230/LIPIcs.ISAAC.2021.6},
annote =	{Keywords: Clustering, Approximation, Hardness}
}```
Document
##### Fast and Exact Convex Hull Simplification

Authors: Georgiy Klimenko and Benjamin Raichel

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

##### Abstract
Given a point set P in the plane, we seek a subset Q ⊆ P, whose convex hull gives a smaller and thus simpler representation of the convex hull of P. Specifically, let cost(Q,P) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q,P) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q,P), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log²n) time algorithm and an O(n log³n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n^2.5302) time algorithm when minimizing k and an O(min{n^2.5302, kn^2.376}) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.

##### Cite as

Georgiy Klimenko and Benjamin Raichel. Fast and Exact Convex Hull Simplification. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

```@InProceedings{klimenko_et_al:LIPIcs.FSTTCS.2021.26,
author =	{Klimenko, Georgiy and Raichel, Benjamin},
title =	{{Fast and Exact Convex Hull Simplification}},
booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
pages =	{26:1--26:17},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-215-0},
ISSN =	{1868-8969},
year =	{2021},
volume =	{213},
editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.26},
URN =		{urn:nbn:de:0030-drops-155373},
doi =		{10.4230/LIPIcs.FSTTCS.2021.26},
annote =	{Keywords: Convex hull, coreset, exact algorithm}
}```
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