Vertex characterization of partition polytopes of bipartitions and of planar point sets

The partition problem concerns the partitioning of n given 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. The problem has applications in diverse 4elds that include clustering, inventory, scheduling and statistical hypothesis testing. Since the objective function is convex, the partition problem can be reduced to the problem of maximizing the same objective over the partition polytope, de4ned to be the convex hull of all solutions. In this article we completely characterize the vertices of partition polytopes when either d=2 or p=2, and determine the maximum number of vertices of any partition polytope of n vectors when d = 2 or p = 2 up to a constant factor. Our characterization implies a bijection between vertices of the polytope of 0-separable partitions, and is best possible in the sense that there are examples in the literature showing the analogue fails already for p = d = 3. Our enumerative results provide lower bounds on the time complexity needed to solve the partition problem when the objective function is presented by an oracle. ? 2002 Elsevier Science B.V. All rights reserved. ∗ Corresponding author. Fax: +972-4-823-5194. E-mail addresses: [email protected] (S. Aviran), [email protected] (S. Onn), [email protected] (U.G. Rothblum). URL: http://ie.technion.ac.il/∼onn 1 Research supported in part by a grant from the Israel Science Foundation. 2 Research supported in part by a grant from the Israel Science Foundation, by a VPR grant at the Technion, and by the Fund for the Promotion of Research at the Technion. 0166-218X/02/$ see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S0166 -218X(01)00326 -2 2 S. Aviran et al. / Discrete Applied Mathematics 124 (2002) 1–15