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Suppose $P$ is a polyhedron whose representation as a system of linear inequalities is given to us: $$ P = \{ x ~|~ Ax \leq b\}$$ Define $P'$ be the image of $P$ under the linear transformation which maps $x$ to $Cx$:
$$ P' = \{ Cx ~|~ x \in P \}.$$ Given $A,b,C$, our goal is to compute a representation of $P'$ as a system of linear inequalities, i.e., to compute $D,e$ satisfying $$ P' = \{ x ~|~ D x \leq e \}.$$ What is the complexity of this problem (i.e., how many arithmetic operations or bit operations are required)?

Let us adopt the convention that $A \in R^{m \times n}$ while $C \in R^{k \times n}$ so the answer should be in terms of $m,n,k$. Further, one may suppose that entries of $P$ are rational numbers whose numerators and denominators take $B$ bits to specify, so that in the bit-model answers should be in terms of $B$ as well.

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    Perhaps you should add something on why you expect this not to be simply the complexity of computing $D=AC^{-1}$ (assuming $C$ is invertible).2012-08-25

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