Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$.

Is there an algorithm which tells you if $X$ has good reduction outside $S$? If yes, does this algorithm run in polynomial time?

I think the answer depends on how one "gives" X. We can give $X$ by explicit equations in some projective space, as a branched cover of $\mathbf{P}^1$ or etc.