Nearest-Neighbor Decompositions of Drawings
Jonas Cleve, Nicolas Grelier, Kristin Knorr, Maarten, L\"offler, Wolfgang Mulzer, Daniel Perz

TL;DR
This paper investigates the computational complexity of decomposing straight-line segment drawings into unions of nearest-neighbor graphs, establishing NP-completeness results, and introduces a conflict graph framework with applications to string and planar graphs.
Contribution
It introduces the conflict graph concept for segment drawings, proves NP-completeness of nearest-neighbor decomposition problems, and provides approximation algorithms and special case analyses.
Findings
NP-complete to decide for c ≥ 3 parts, even without crossings
Polynomial-time for c=2 in non-crossing segments
Conflict graphs include string and planar graphs
Abstract
Let be a set of straight-line segments in the plane, potentially crossing, and let be a positive integer. We denote by the union of the endpoints of the straight-line segments of and of the intersection points between pairs of segments. We say that has a nearest-neighbor decomposition into parts if we can partition into point sets such that is the union of the nearest neighbor graphs on . We show that it is NP-complete to decide whether can be drawn as the union of nearest-neighbor graphs, even when no two segments cross. We show that for , it is NP-complete in the general setting and polynomial-time solvable when no two segments cross. We show the existence of an -approximation algorithm running in subexponential time for…
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Remote Sensing and LiDAR Applications · 3D Modeling in Geospatial Applications
