Lately, I saw an NP-complete proof that involves creating an instance of a problem using $\infty$. Is this a polynomial-time reduction?
More precisely, let a problem $\Pi$ has an instance $I=(n, A, v)$ where $A$ is a matrix of size $n\times n$, $A=[a_{ij}]$, and $v$ is positive.
The proof that $\Pi$ is NP-complete works by reducing the independent set problem (an instance is a graph $G=(V,E)$ and an integer $k$) to $\Pi$ as follows: $n=|V|,v=k$ and
$$a_{ij} = \begin{cases} \infty \text{ if } \{i,j\}\in E\\ 0, \text{ otherwise } \end{cases}.$$ Also, $G$ has an independent set of size $k\iff\Pi$ has a value $v= k$.
I do not see how one can set $a_{ij}$ to $\infty$.