Suppose that there are $M$ left nodes, and $N$ right nodes. Suppose that the $n$th right node can connect $K_n$ left nodes, $n=1,\cdots,N$. Let $\Lambda_m$ denote the set of right nodes that is connected to the $m$th left node, $m=1,\cdots,M$. If the $m$th left node is connected to the $n$th right node, i.e., $n\in \Lambda_m$, then there is no penalty; otherwise, i.e., $n\notin \Lambda_m$, there will be a panelty $K_n$ to the $m$th left node for not connecting to the $n$th right node, i.e., the capacity of the $n$th right node.
Then, given any assignment $\Lambda_m$'s, the total panelty of the $m$th left node for not connecting the right nodes is
\begin{align} P_m=\sum\limits_{n\notin \Lambda_m} K_n, ~~~ m=1,\cdots,M. \end{align}
The problem is to find the optimal assignment $\Lambda_m$'s to minimize the maximum penalty among all the $M$ left nodes, i.e.,
\begin{align} {\rm minimize} ~ \max(P_1,\cdots,P_M), \end{align}suppose that each right node $n$ can only connect $K_n$ left nodes.
Is there any efficient way to solve this problem?