Momentopes , the Complexity of Vector Partitioning , and

The computational complexity of the partition problem, which concerns the partitioning of a set of n vectors in d-space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part, is determined by the number of vertices of the corresponding p-partition polytope defined to be the convex hull in (d × p)-space of all solutions. In this article, introducing and using the class of Momentopes, we establish the lower bound v p,d (n) = (n (d−1)/2 p) on the maximum number of vertices of any p-partition polytope of a set of n points in d-space, which is quite compatible with the recent upper bound v p,d (n) = O(n d(p−1)−1), implying the same bound on the complexity of the partition problem. We also discuss related problems on the realizability of Davenport–Schinzel sequences and describe some further properties of Momentopes. The partition problem concerns the partitioning of a set A of n vectors in d-space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part. More precisely, with each p-partition of A, namely, an ordered *