A Polynomial Time Algorithm for Shaped Partition Problems

We consider the class of Shaped Partition Problems of partitioning n given vectors in d-dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. This class has broad expressive power and captures NP-hard problems even if either d or p is xed. In contrast, we show that when both d and p are xed, the problem can be solved in strongly polynomial time. Our solution method relies on studying the corresponding class of Shaped Partition Polytopes. Such polytopes may have exponentially many vertices and facets even when one of d or p is xed; but, we show that when both d and p are xed, the number of vertices of any Shaped Partition Polytope is O(n d(p 2)) and all vertices can be produced in strongly polynomial time.