An Optimal Algorithm for Cardinality-Constrained Diameter Partitioning

TL;DR

This paper presents an optimal quadratic-time algorithm for cardinality-constrained diameter partitioning, improving previous methods and applying to Euclidean weights with subquadratic solutions in fixed dimensions.

Contribution

It introduces an $O(n^2)$ algorithm for the problem, matching lower bounds, and extends to Euclidean weights with faster solutions in fixed dimensions.

Findings

The algorithm computes optimal partitions for all cardinalities simultaneously.

It reduces the problem to a bottleneck 2-coloring on the maximum spanning tree.

For Euclidean weights, a subquadratic algorithm is achieved in fixed dimensions.

Abstract

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.