Augmenting a Geometric Matching is $NP$-complete
Tillmann Miltzow
TL;DR
This paper proves that determining whether a partial geometric matching can be extended to a perfect non-crossing matching is NP-complete, highlighting computational complexity in geometric graph augmentation.
Contribution
It establishes NP-completeness of augmenting partial geometric matchings to perfect matchings, even in bichromatic cases, via a reduction from 1-in-3-SAT.
Findings
Deciding augmentability is NP-complete.
The NP-completeness holds for bichromatic matchings.
The result connects geometric matching problems with classical NP-complete problems.
Abstract
Given points in the plane, it is well-known that there always exists a perfect straight-line non-crossing matching. We show that it is -complete to decide if a partial matching can be augmented to a perfect one, via a reduction from 1-in-3-SAT. This result also holds for bichromatic matchings.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Data Management and Algorithms · 3D Shape Modeling and Analysis