Computing simple circuits from a set of line segments is NP-complete
SCG '87 Proceedings of the third annual symposium on Computational geometry
Noncrossing subgraphs in topological layouts
SIAM Journal on Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Bottleneck non-crossing matching in the plane
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Bottleneck non-crossing matching in the plane
Computational Geometry: Theory and Applications
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We study some geometric maximization problems in theEuclidean plane under the non-crossing constraint. Given a setV of2n points in general position in theplane, we investigate the following geometric configurations usingstraight-line segments and the Euclidean norm: (i) longest non-crossingmatching, (ii) longest non-crossing hamiltonian path, (iii) longestnon-crossing spanning tree. We propose simple and efficient algorithmsto approximate these structures within a constant factor of optimality.Somewhat surprisingly, we also show that our bounds are within aconstant factor of optimality even without the non-crossing constraint.For instance, we give an algorithm to compute a non-crossing matchingwhose total length is at least 2/&pgr; of the longest (possiblycrossing) matching, and show that the ratio 2/&pgr; between thenon-crossing and crossing matching is the best possible. Perhaps due totheir utter simplicity, our methods also seem more general and amenableto applications in other similar contexts.