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 worstcase 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 ddimensional hypercubes we maintain a (1+ε)2^dapproximate solution in the unweighted case and a O(2^d)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 ddimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1+ε)log^{d1}N.
BibTeX  Entry
@InProceedings{henzinger_et_al:LIPIcs:2020:12209,
author = {Monika Henzinger and Stefan Neumann and Andreas Wiese},
title = {{Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {51:151:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771436},
ISSN = {18688969},
year = {2020},
volume = {164},
editor = {Sergio Cabello and Danny Z. Chen},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12209},
URN = {urn:nbn:de:0030drops122094},
doi = {10.4230/LIPIcs.SoCG.2020.51},
annote = {Keywords: Dynamic algorithms, independent set, approximation algorithms, interval graphs, geometric intersection graphs}
}
Keywords: 

Dynamic algorithms, independent set, approximation algorithms, interval graphs, geometric intersection graphs 
Collection: 

36th International Symposium on Computational Geometry (SoCG 2020) 
Issue Date: 

2020 
Date of publication: 

08.06.2020 