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 clusterdistances. Furthermore, we investigate clustering of planar point sets into convex quadrilaterals.
Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems
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