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I have a reduction that proves that a problem is NP-hard to approximate to a factor $1 + \epsilon$ for any $0 < \epsilon < 1$. The reduction is polynomial in $n$ (the size of the instance of the problem I'm reducing from). However, the reduction also uses $1/\epsilon$. Since $\epsilon$ is a constant, this is considered a valid reduction.

What I'm concerned about is that as $\epsilon$ tends to zero, $1/\epsilon$ tends to infinity. How is my reduction a valid polynomial time reduction if it uses $1/\epsilon$ as part of the reduction construction when $1/\epsilon$ can tend to infinity. Most reductions have assumed this is valid since $1/\epsilon$ is a constant since $\epsilon$ is a constant. But I would like a more intuitive explanation of why this is still valid.

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You can think of $\epsilon$ as parametrizing a family of reductions to a family of approximation problems:

  • The $\epsilon$-th problem is to compute a $1+\epsilon$ approximation to the underlying optimization problem.

  • The $\epsilon$-th reduction shows that the $\epsilon$-th problem is $\mathrm{NP}$-hard.

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