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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.52
URN: urn:nbn:de:0030-drops-160609
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16060/
Kumar, Neeraj ;
Lokshtanov, Daniel ;
Saurabh, Saket ;
Suri, Subhash ;
Xue, Jie
Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles
Abstract
Suppose we are given a pair of points s, t and a set ? of n geometric objects in the plane, called obstacles. We show that in polynomial time one can construct an auxiliary (multi-)graph G with vertex set ? and every edge labeled from {0, 1}, such that a set ?_d ⊆ ? of obstacles separates s from t if and only if G[?_d] contains a cycle whose sum of labels is odd. Using this structural characterization of separating sets of obstacles we obtain the following algorithmic results.
In the Obstacle-removal problem the task is to find a curve in the plane connecting s to t intersecting at most q obstacles. We give a 2.3146^q n^{O(1)} algorithm for Obstacle-removal, significantly improving upon the previously best known q^{O(q³)} n^{O(1)} algorithm of Eiben and Lokshtanov (SoCG'20). We also obtain an alternative proof of a constant factor approximation algorithm for Obstacle-removal, substantially simplifying the arguments of Kumar et al. (SODA'21).
In the Generalized Points-separation problem input consists of the set ? of obstacles, a point set A of k points and p pairs (s₁, t₁), … (s_p, t_p) of points from A. The task is to find a minimum subset ?_r ⊆ ? such that for every i, every curve from s_i to t_i intersects at least one obstacle in ?_r. We obtain 2^{O(p)} n^{O(k)}-time algorithm for Generalized Points-separation. This resolves an open problem of Cabello and Giannopoulos (SoCG'13), who asked about the existence of such an algorithm for the special case where (s₁, t₁), … (s_p, t_p) contains all the pairs of points in A. Finally, we improve the running time of our algorithm to f(p,k) ⋅ n^{O(√k)} when the obstacles are unit disks, where f(p,k) = 2^{O(p)} k^{O(k)}, and show that, assuming the Exponential Time Hypothesis (ETH), the running time dependence on k of our algorithms is essentially optimal.
BibTeX - Entry
@InProceedings{kumar_et_al:LIPIcs.SoCG.2022.52,
author = {Kumar, Neeraj and Lokshtanov, Daniel and Saurabh, Saket and Suri, Subhash and Xue, Jie},
title = {{Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16060},
URN = {urn:nbn:de:0030-drops-160609},
doi = {10.4230/LIPIcs.SoCG.2022.52},
annote = {Keywords: points-separation, min color path, constraint removal, barrier resillience}
}
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Keywords: |
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points-separation, min color path, constraint removal, barrier resillience |
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Collection: |
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38th International Symposium on Computational Geometry (SoCG 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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01.06.2022 |
