Is the function `maximum_weight_maximal_matching`correct?

[simonschoelly]

I was looking at this, because I wanted to convert it to Julia v1.0, but I'm not sure this function actually does what it should do.
From my understanding, it tries to solve the following (simplified here) linear program:

Given bipartite graph G with parts |V1| <= |V2|  and weightmap W = {w_ij}

maximize  Sum( x_ij * w_ij  ) where   w_ij > 0   s.t.
(I)   forall i,j     :  x_ij >= 0
(II)  forall i in V1 : Sum(x_ij) == 1 where j in neighbors(G, i)
(III) forall j in V2 : Sum(x_ij) <= 1 where i in neighbors(G, j)

There are two issues:

1. Now the objective function clearly aims to maximize the total weight. But I don't see why it only does this among the matchings with the largest number of edges. And shouldn't it be called maximum_weight_maximum_matching then? After all, the documentation suggests it looks for a maximum set of edges and not simply a maximal one.

2. Condition (II) is not possible in all cases (at least for integer solutions). For example there are bipartite graphs with parts of equal size that do not contain a perfect matching, even if the graph is connected and does not have isolated vertices.

[Wikunia]

Regarding to 1. I'm definitely not an expert but yes maximum_weight_maximum_matching sounds correct for me.
Regarding 2. I just read: " If the graph is not complete bipartite, missing edges are inserted with value zero." here https://en.wikipedia.org/wiki/Matching_(graph_theory) which means that there exists a maximum matching in this case. But the algorithm doesn't do this which I also think is reasonable somehow as it can be currently used for maximum_matching if we have weights set to 1.