Approximation Algorithms for the Minimum Convex Partition Problem

Abstract

We present two algorithms that compute constant factor approximations of a minimum convex partition of a set P of n points in the plane. The first algorithm is very simple and computes a 3-approximation in O(n logn) time. The second algorithm improves the approximation factor to $\frac{30}{11} < 2.7273$ but it is more complex and a straight forward implementation will run in O(n ²) time. The claimed approximation factors are proved under the assumption that no three points in P are collinear. As a byproduct we obtain an improved combinatorial bound: there is always a convex partition of P with at most $\frac{15}{11}n -- \frac{24}{11}$ convex regions.