Chebyshev Center of the Intersection of Balls: Complexity, Relaxation and Approximation

TL;DR

This paper investigates the computational complexity of finding the smallest enclosing ball for intersecting balls, proving NP-hardness in general, but also identifying special cases where polynomial solutions are possible and providing approximation bounds.

Contribution

It establishes the NP-hardness of the Chebyshev center problem for intersecting balls and introduces a linear programming relaxation to derive approximation bounds and polynomial solvability in special cases.

Findings

Proves NP-hardness of the problem when p > n.

Provides a linear programming relaxation with approximation bounds.

Shows polynomial solvability when n or p - n is fixed.

Abstract

We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem (CCB). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem (UQ). When p<=n, (UQ) is known to enjoy a strong duality and consequently (CCB) is solved via a standard convex quadratic programming (SQP). In this paper, we first prove that (CCB) is NP-hard and the special case when n = 2 is strongly polynomially solved. With the help of a newly introduced linear programming relaxation (LP), the (SQP) relaxation is reobtained more directly and the first approximation bound for the solution obtained by (SQP) is established for the hard case p>n. Finally, also based on (LP), we show that (CCB) is polynomially solved when either n or p-n(> 0) is fixed.