2 Search Results for "Bhattiprolu, Vijay V. S. P."

Document
DOI: 10.4230/LIPIcs.SoCG.2016.18
Separating a Voronoi Diagram via Local Search

Authors: Vijay V. S. P. Bhattiprolu and Sariel Har-Peled

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
Given a set P of n points in R^d , we show how to insert a set Z of O(n^(1-1/d)) additional points, such that P can be broken into two sets P1 and P2 , of roughly equal size, such that in the Voronoi diagram V(P u Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1,P2) of P , we present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition.
Cite as

Vijay V. S. P. Bhattiprolu and Sariel Har-Peled. Separating a Voronoi Diagram via Local Search. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bhattiprolu_et_al:LIPIcs.SoCG.2016.18,
  author =	{Bhattiprolu, Vijay V. S. P. and Har-Peled, Sariel},
  title =	{{Separating a Voronoi Diagram via Local Search}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.18},
  URN =		{urn:nbn:de:0030-drops-59107},
  doi =		{10.4230/LIPIcs.SoCG.2016.18},
  annote =	{Keywords: Separators, Local search, Approximation, Voronoi diagrams, Delaunay triangulation, Meshing, Geometric hitting set}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.152
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

Authors: Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract
A hypergraph is said to be X-colorable if its vertices can be colored with X colors so that no hyperedge is monochromatic. 2-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 2-colorable k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer than a fraction 2^(-k+1) of hyperedges (which is trivially achieved by a random 2-coloring), and the best algorithms to color the hypergraph properly require about n^(1-1/k) colors, approaching the trivial bound of n as k increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 2-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 2-colorability: (A) Low-discrepancy: If the hypergraph has a 2-coloring of discrepancy l << sqrt(k), we give an algorithm to color the hypergraph with about n^(O(l^2/k)) colors. However, for the maximization version, we prove NP-hardness of finding a 2-coloring miscoloring a smaller than 2^(-O(k)) (resp. k^(-O(k))) fraction of the hyperedges when l = O(log k) (resp. l=2). Assuming the Unique Games conjecture, we improve the latter hardness factor to 2^(-O(k)) for almost discrepancy-1 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (k-l)-coloring such that each hyperedge is polychromatic with all these colors (this is stronger than a (l+1)-discrepancy 2-coloring), we give a 2-coloring algorithm that miscolors at most k^(-Omega(k)) of the hyperedges when l << sqrt(k), and complement this with a matching Unique Games hardness result showing that when l = sqrt(k), it is hard to even beat the 2^(-k+1) bound achieved by a random coloring. (C) Strong Colorability: We obtain similar (stronger) Min- and Max-2-Coloring algorithmic results in the case of (k+l)-strong colorability.
Cite as

Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Approximate Hypergraph Coloring under Low-discrepancy and Related Promises. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 152-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bhattiprolu_et_al:LIPIcs.APPROX-RANDOM.2015.152,
  author =	{Bhattiprolu, Vijay V. S. P. and Guruswami, Venkatesan and Lee, Euiwoong},
  title =	{{Approximate Hypergraph Coloring under Low-discrepancy and Related Promises}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{152--174},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.152},
  URN =		{urn:nbn:de:0030-drops-53011},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.152},
  annote =	{Keywords: Hypergraph Coloring, Discrepancy, Rainbow Coloring, Stong Coloring, Algorithms, Semidefinite Programming, Hardness of Approximation}
}
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