Abstract
We revisit several maximization problems for geometric networks design
under the noncrossing constraint, first studied by Alon, Rajagopalan
and Suri (ACM Symposium on Computational Geometry, 1993).
Given a set of $n$ points in the plane in general position (no three points collinear), compute a longest noncrossing configuration composed of straight line segments that is: (a) a matching (b) a Hamiltonian path (c) a spanning tree. Here we obtain new results for (b) and (c), as well as for the Hamiltonian cycle problem:
(i) For the longest noncrossing Hamiltonian path problem,
we give an approximation algorithm with ratio $\frac{2}{\pi+1} \approx 0.4829$. The previous best ratio, due to Alon et al., was $1/\pi \approx 0.3183$. Moreover, the ratio of our algorithm is close to $2/\pi$ on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time.
(ii) For the longest noncrossing spanning tree problem, we give an
approximation algorithm with ratio $0.502$ which runs in $O(n \log{n})$ time. The previous ratio, $1/2$, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first rederive the result of Alon et al. with a faster $O(n \log{n})$time algorithm and a very simple analysis.
(iii) For the longest noncrossing Hamiltonian cycle problem,
we give an approximation algorithm whose ratio is close to $2/\pi$ on a relatively broad class of instances: for point sets with the product
$\bf{\langle}$~diameter~$\times$ ~convex hull size $\bf{\rangle}$ much smaller than the maximum length matching. The algorithm runs in
$O(n^{7/3}\log{n})$ time. No previous approximation results
were known for this problem.
BibTeX  Entry
@InProceedings{dumitrescu_et_al:LIPIcs:2010:2465,
author = {Adrian Dumitrescu and Csaba D. T{\'o}th},
title = {{Long Noncrossing Configurations in the Plane}},
booktitle = {27th International Symposium on Theoretical Aspects of Computer Science},
pages = {311322},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897163},
ISSN = {18688969},
year = {2010},
volume = {5},
editor = {JeanYves Marion and Thomas Schwentick},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2465},
URN = {urn:nbn:de:0030drops24655},
doi = {10.4230/LIPIcs.STACS.2010.2465},
annote = {Keywords: Longest noncrossing Hamiltonian path, longest noncrossing Hamiltonian cycle, longest noncrossing spanning tree, approximation algorithm.}
}
Keywords: 

Longest noncrossing Hamiltonian path, longest noncrossing Hamiltonian cycle, longest noncrossing spanning tree, approximation algorithm. 
Seminar: 

27th International Symposium on Theoretical Aspects of Computer Science 
Issue Date: 

2010 
Date of publication: 

09.03.2010 