Minimize longest continuous streak of $1$s after flipping $m$ $-1$s to $1$s
You can use dynamic programming. It is very likely that much faster algorithms exist.

Given a string $w$ of length $n$, for $0 \leq i,j \leq n$ and $0 \leq k \leq m$ you compute the minimal maximum 1-run in $w_1,\ldots,w_i$ given that $k$ many -1s were changed to 1s, and given that $w_1,\ldots,w_i$ ends in a 1-run of length $j$.

This gives an $O(n^2m) = O(n^3)$ algorithm.