Computing simple circuits from a set of line segments is NP-complete
SCG '87 Proceedings of the third annual symposium on Computational geometry
Disjoint simplices and geometric hypergraphs
Proceedings of the third international conference on Combinatorial mathematics
Noncrossing subgraphs in topological layouts
SIAM Journal on Discrete Mathematics
Reconstructing sets of orthogonal line segments in the plane
Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
An upper bound on the number of planar k-sets
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Computational geometry column 54
ACM SIGACT News
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We study some geometric maximization problems in the Euclidean plane under the non-crossing constraint. Given a set V of 2n points in general position in the plane, we investigate the following geometric configurations using straight-line segments and the Euclidean norm: (i) longest non-crossing matching, (ii) longest non-crossing hamiltonian path, (iii) longest non-crossing spanning tree. We propose simple and efficient algorithms to approximate these structures within a constant factor of optimality. Somewhat surprisingly, we also show that our bounds are within a constant factor of optimality even without the non-crossing constraint, For instance, we give an algorithm to compute a non-crossing matching whose total length is at least 2/π of the longest (possibly crossing) matching, and show that the ratio 2/π between the non-crossing and crossing matching is the best possible. Perhaps due to their utter simplicity, our methods also seem more general and amenable to applications in other similar contexts.