I would like to know if the following problem is known and has any efficient solution.
Given an $n\times n$ score matrix $S$. Find the best $a$ elements, in terms of their sum of scores, such that no row or column is selected more than $b$ times.
This can be re-formulated as the following integer linear program: \begin{align} \mbox{maximize} &\sum_{1\le i,j\le n} s_{ij}x_{ij},\\ \mbox{subject to} & \sum_{1\le i,j\le n} x_{ij} = a,\\ & \sum_{1\le i\le n} x_{ij} \le b \quad\forall 1\le j\le n,\\ & \sum_{1\le j\le n} x_{ij} \le b \quad\forall 1\le i\le n,\\ & x_{ij}\in\{0,1\}\quad\forall 1\le i,j\le n. \end{align}
Thank you very much in advance for your help!