I wish to take a look at online/approximate weighted and capacitated bipartite matching problem.Consider $G=\{L\cup R, E\}$, $|L|=n_1$, $|R|=n_2$, $|E|=m$ and $E\subseteq L\times R$. For each $r_i\in R$, it has capacity $c_i$ which means that at most $c_i$ nodes from $L$ can be matched to $r_i$. The objective function to maximize is $\sum_{i=1}^{n_2}x_iw_i$ where $x_i$ is the number of nodes in $L$ matched to $r_i$ and $w_i>0$ is the weight. The constraints are (1) $x_i\in\{0,...,c_i\}$, (2) any node in $L$ can be matched at most once and (3) any node $l_j$ is allowed to be matched to $r_i$ if $(l_j, r_i)$  $\in E$.Is there any paper that solved the exact problem as I described above (provides either approximate or online algorithm)? To be clear, I am asking for references, and methods are not necessary.
Online/approximate weighted and capacitated bipartite matching
graph theory;approximation algorithms;matching;online algorithms;bipartite graphs
I think there is no paper solving that exact problem, but Online Vertex-Weighted Matching by Aggarwal, Goel, Karande, and Mehta (2011) is very close. If I understood correctly, they solve your problem only with all capacities equal to one.My best guess is that you will have to do some work to extend their guarantees and algorithm to your setting. On the other hand, a greedy algorithm probably should guarantee 0.5 of optimal by a standard argument: Consider the opt matching; for every slot in OPT left unmatched by greedy, there was some other match made that was more valuable (since the slot was available, its partner must have gone elsewhere). I hope you will forgive this discussion of methods contrary to your request.(edit: I want to mention that this argument wouldn't work if the weights are on the arriving vertices; it's good to think about why.)