shortest form $s$ to $t$ stopping at $u$
What you said is true.

Using your notations:

$P$ : path from $s$ to $t$

$M$ : path from $s$ to $u$

$K$ : path from $u$ to $t$

Notice that $P,M,K$ are all shortest paths from their respective sources to their targets.

Following the logic above:

$M \cup K$ is the shortest path from $s$ to $u$ to $t$

hence if $|M \cup K|$ is a good enough result for you - accept it. If it's not, i.e $|M \cup K| &gt; \alpha|P|$, then essentially no shorter path exists from $s$ to $t$ using $u$.

(you can try proving $|M \cup K| \geq |P|$, may help your intuition!