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arXiv · 2303.00147

Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

David Eppstein
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Abstract

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

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David Eppstein. 2023-03-01. Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time. https://doi.org/10.1007/s00453-024-01255-y

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