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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 $$pu003en$$. Finally, also based on (LP), we show that ($${\mathrm{CC}}_{\mathrm{B}}$$) is polynomially solvable when either n or $$p-n(u003e0)$$ is fixed.

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