Reconstruction of weakly simple polygons from their edges

TL;DR

This paper investigates the computational complexity of reconstructing weakly simple polygons from given line segments, establishing NP-completeness in general and providing efficient algorithms for special cases, along with bounds for related subdivision and augmentation problems.

Contribution

It proves NP-completeness for recognizing weakly simple polygons and offers linear-time algorithms for specific cases, extending the understanding of polygon reconstruction problems.

Findings

NP-complete for general weakly simple polygon recognition

Linear-time algorithms for collinear segments and general position cases

Bounds for subdivision and augmentation problems to form weakly simple polygons

Abstract

Given $n$ line segments in the plane, do they form the edge set of a \emph{weakly simple polygon}; that is, can the segment endpoints be perturbed by at most $\varepsilon$, for any $\varepsilon>0$, to obtain a simple polygon? While the analogous question for \emph{simple polygons} can easily be answered in $O(n\log n)$ time, we show that it is NP-complete for weakly simple polygons. We give $O(n)$-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are \emph{directed}, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the $n$ input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.