A Bound on the Sum of Weighted Pairwise Distances of Points Constrained to Balls

TL;DR

This paper establishes a tight upper bound on the sum of weighted pairwise distances for points constrained within balls, providing a dual formulation and a polynomial-time approximation method.

Contribution

It introduces a dual minimization problem for the constrained distance sum and proves strong duality under certain conditions, along with a near-optimal algorithm.

Findings

Strong duality holds when points are affinely independent.

Derived a tight upper bound for the sum of distances.

Proposed a polynomial-time algorithm for near-optimal solutions.

Abstract

We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality holds (i.e., the resulting upper bound is tight) when some locally optimal configuration of points is affinely independent. We sketch a polynomial time algorithm for finding a near-optimal set of points.