I'm watching this video of an example of the Augmenting Path Algorithm. https://youtu.be/C9c8zEZXboA?t=240
For convenience, I name the vertices of X, from left to right, $x_1, x_2, x_3, x_4, x_5, x_6$,respectively, and similarly, the vertices of Y, $y_1, y_2,y_3,y_4,y_5,y_6$.
At the 4' mark, the presenter found an augmenting path, namely $y_3x_2y_4x_5$. However, I notice that there is also another augmenting path that also originates from $y_3$ and was not chosen, namely $y_3x_2y_4x_4y_2x_1$. Is there any reason why the former path is preferred?
For your information, here's the algorithm, and in particular the itereation step: https://youtu.be/ory4WMX0rDU?t=306
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Edit: my other question concerning the correctness of the algorithm. The algorithm outputs $T \cup (X-R)$ as a vertex cover of the graph. I can see that $T$ is in there because it "takes care" of the vertices of $R$ in $X$. Adding $(X-R)$ to the set obviously includes all the remaining vertices of $X$ in the cover, but how can we be sure that the neighbors of $(X-R)$ are all of $(Y-T)$?
Edit 2: I've just found an example for the problem above.
The red edges are a maximum matching of the graph, which give us $T = \{y_3,y_5\}$, and $(X - R) = \{x_2\}$. Clearly $T \cup (X-R)$ leaves out $y_1,y_2$. Why isn't this case mentioned anywhere in the algorithm?