An Optimal Algorithm for Cardinality-Constrained Diameter Partitioning

2026-05-05

AI Summary

This study addresses the problem of partitioning a set of $n$ elements into two subsets of prescribed sizes so as to minimize the larger of their diameters. By reformulating the problem as a bottleneck 2-coloring on a maximum spanning tree, the authors combine tree-based dynamic programming with geometric algorithmic techniques to present the first optimal algorithm running in $O(n^2)$ time. They further establish an $\Omega(n^2)$ query lower bound in the comparison model where only pairwise distance queries are allowed. For the special case of equal-sized subsets in Euclidean space of fixed dimension, they devise a subquadratic-time algorithm that significantly improves upon Avis’s $O(n^2 \log n)$ approach.

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 $Ω(n^2)$ 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.

Problem

• cardinality-constrained
• diameter partitioning
• optimal algorithm
• bottleneck 2-coloring
• Euclidean weights

Innovation

• diameter partitioning
• cardinality constraint
• bottleneck 2-coloring
• maximum spanning tree
• subquadratic algorithm