An Optimal Algorithm for Cardinality-Constrained Diameter Partitioning

Cardinality-constrained diameter partitioning asks for a partition of $n$ items into two classes of prescribed sizes that minimizes the larger of the two class diameters. We give an $O(n^2)$ algorithm and a matching $\Omega(n^2)$ lower bound if we can only query the weight between two elements. The algorithm computes the optimum for every cardinality simultaneously, improving Avis's $O(n^2\log n)$. The reduction is to a bottleneck 2-coloring problem on the maximum spanning tree, solved by a standard tree DP. For a single cardinality with Euclidean weights, we obtain a subquadratic time algorithm in any fixed dimension.