Discrete and Computational Geometry

This talk surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized. This problem is known to be NP-complete even for k = 3. The case of k = 2, that is, bipartition problem is solvable in polynomial time. On the other hand, in geometric setting where vertices are points in the plane and weights of edges equal the distances between corresponding points, the same problem is solvable in polynomial time even for k ≥ 3 as far as k is a fixed constant. For the case k = 2, effective use of geometric property of an optimal solution leads to considerable improvement on the computational complexity. Other related topics are also discussed.