Triangle Covers of Polygons: Approximation Algorithms and Complexity

Abstract

We study the geometric set cover problems of covering and partitioning a polygon into a minimum number of triangles. While the decision version of the covering problem is known to be APX-hard and $\exists\mathbb R$-complete (even for polygons without holes and vertices in general position), the computational complexity of partitioning a simple polygon into a minimum number of triangles is open. Our main result refutes a 40 year old conjecture by Asano, Asano, and Pinter stating that the triangle vertices of a minimum partition lie on intersection points of special lines, which are iteratively generated by the polygon vertices and the intersection points. In particular, we show that there exist polygons where all minimum triangle partitions need Steiner points with irrational coordinates. This indicates that the problem might be more complex than previously expected. Complementarily, we provide a simple constant factor approximation to compute a minimum triangle partition of a given polygon (possibly with holes). In the context of coverings, we devise the first constant factor approximation for computing a minimum triangle cover of a simple polygon. For general polygons, an $O(\log n)$-approximation follows by close inspection of previous work.