using the ellipsoid algorithm to find a poly time algorithm for the optimization problem

Question

Consider the following optimization problem: Let $n$ be even and let $c, x$ be positive vectors in $\mathbb{R}^n.$ Find $\min(c^Tx)$ for $\sum_S x_i\geq 1,$ for any $S\subset \{1,...,n\}$ with $|S| =n/2$. We have to show that the ellipsoid algorithm can be used to get a poly(n)-time algorithm for the problem.