Some Geometric Clustering Problems

This paper investigates the computational complexity of several clustering problems with special objective functions for point sets in the Euclidean plane. Our strongest negative result is that clustering a set of 3k points in the plane into k triangles with minimum total circumference is NP-hard. On the other hand, we identify several special cases that are solvable in polynomial time due to the special structure of their optimal solutions: The clustering of points on a convex hull into triangles; the clustering into equal-sized subsets of points on a line or on a circle with special objective functions; the clustering with minimal cluster-distances. Furthermore, we investigate clustering of planar point sets into convex quadrilaterals.