On Maximizing the Distance to a Given Point over an Intersection of Balls II

TL;DR

This paper investigates the problem of maximizing the distance to a fixed point over an intersection of balls, proving a conjecture that enables polynomial algorithms for certain points inside the convex hull of the ball centers.

Contribution

It proves a conjecture that extends polynomial solvability to points on the facets of the convex hull, including interior points, under specific conditions.

Findings

Polynomial algorithm exists for points on the convex hull facets.

The conjecture is proved under slightly stronger conditions.

Points in a small ball share the maximizer with the facet points.

Abstract

In this paper the problem of maximizing the distance to a given fixed point over an intersection of balls is considered. It is known that this problem is NP complete in the general case, since any subset sum problem can be solved upon solving a maximization of the distance over an intersection of balls to a point inside the convex hull. The general context is: in [1] it is shown that exists a polynomial algorithm which always solves the maximization problem if the given point is outside the convex hull of the centers of the balls. Naturally one asks if there is a polynomial algorithm which solves the problem for a point inside the convex hull. A conjecture stated in a previous paper, [1] is proved, under slightly stronger conditions. The proven conjecture allows a polynomial algorithm for points on the facets of the convex hull and shows that such points share the maximizer with all the points in a small enough ball centered at it, thus including points in the interior of the convex hull of the ball centers.