Abstract
Independent set is a fundamental problem in combinatorial optimization. While in general graphs the problem is essentially inapproximable, for many important graph classes there are approximation algorithms known in the offline setting. These graph classes include interval graphs and geometric intersection graphs, where vertices correspond to intervals/geometric objects and an edge indicates that the two corresponding objects intersect. We present dynamic approximation algorithms for independent set of intervals, hypercubes and hyperrectangles in d dimensions. They work in the fully dynamic model where each update inserts or deletes a geometric object. All our algorithms are deterministic and have worst-case update times that are polylogarithmic for constant d and ε > 0, assuming that the coordinates of all input objects are in [0, N]d and each of their edges has length at least 1. We obtain the following results: For weighted intervals, we maintain a (1 + ε)-approximate solution. For d-dimensional hypercubes we maintain a (1 + ε)2d-approximate solution in the unweighted case and a O(2d)-approximate solution in the weighted case. Also, we show that for maintaining an unweighted (1 + ε)-approximate solution one needs polynomial update time for d ≥ 2 if the ETH holds. For weighted d-dimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1 + ε) logd−1 N.
Original language | English |
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Title of host publication | 36th International Symposium on Computational Geometry, SoCG 2020 |
Editors | Sergio Cabello, Danny Z. Chen |
Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
ISBN (Electronic) | 9783959771436 |
DOIs | |
Publication status | Published - 1 Jun 2020 |
Event | 36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland Duration: 23 Jun 2020 → 26 Jun 2020 |
Publication series
Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 164 |
ISSN (Print) | 1868-8969 |
Conference
Conference | 36th International Symposium on Computational Geometry, SoCG 2020 |
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Country/Territory | Switzerland |
City | Zurich |
Period | 23/06/20 → 26/06/20 |
Bibliographical note
Funding Information:Funding Monika Henzinger: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 340506. Stefan Neumann: Part of this work was done while visiting Brown University. Stefan Neumann gratefully acknowledges the financial support from the Doctoral Programme “Vienna Graduate School on Computational Optimization” which is funded by the Austrian Science Fund (FWF, project no. W1260-N35). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 340506. Andreas Wiese: Andreas Wiese was supported by the grant Fondecyt Regular 1170223.
Publisher Copyright:
© Monika Henzinger, Stefan Neumann, and Andreas Wiese; licensed under Creative Commons License CC-BY 36th International Symposium on Computational Geometry (SoCG 2020).
Funding
Funding Monika Henzinger: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 340506. Stefan Neumann: Part of this work was done while visiting Brown University. Stefan Neumann gratefully acknowledges the financial support from the Doctoral Programme “Vienna Graduate School on Computational Optimization” which is funded by the Austrian Science Fund (FWF, project no. W1260-N35). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 340506. Andreas Wiese: Andreas Wiese was supported by the grant Fondecyt Regular 1170223.
Keywords
- Approximation algorithms
- Dynamic algorithms
- Geometric intersection graphs
- Independent set
- Interval graphs