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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.

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I tried to edit your question, but I can't really see what you mean to say about the sets $S$. –  Kevin Carlson Oct 22 '12 at 10:42
    
yeah thanks for the editing.S are just the set of indices –  elnino Oct 22 '12 at 10:49
    
also S is a subset of [n] not belongs to...[n] represents the set of n linearly independent columns –  elnino Oct 22 '12 at 10:52
    
Do you mean that whenever I take $n/2$ elements of $x$, they sum to at least $1$? –  Kevin Carlson Oct 22 '12 at 11:04
    
yeah...right...xi's are elements of x.So x1,x2...xn/2 –  elnino Oct 22 '12 at 11:32
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