Chebyshev center of the intersection of balls: complexity, relaxation and approximation

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 (${\mathrm{CC}}_{\mathrm{B}}$). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem ($\mathrm{UQ}$). When $p\le n$, ($\mathrm{UQ}$) is known to enjoy a strong duality and consequently (${\mathrm{CC}}_{\mathrm{B}}$) is solved via a standard convex quadratic programming ($\mathrm{SQP}$). In this paper, we first prove that (${\mathrm{CC}}_{\mathrm{B}}$) is NP-hard and the special case when $n=2$ is polynomially solvable. With the help of a newly introduced linear programming relaxation (LP), the ($\mathrm{SQP}$) relaxation is reobtained more directly and the first approximation bound for the solution obtained by ($\mathrm{SQP}$) is established for the hard case $p>n$. Finally, also based on (LP), we show that (${\mathrm{CC}}_{\mathrm{B}}$) is polynomially solvable when either n or $p-n(>0)$ is fixed.