Maximum stable matching/allocation I checked some papers on two-side stable allocation/matching (marriage, worker/company, doctor/hospital), but has not found any literature on the following problem. Can someone point out if I missed some paper or help analyze the problem?

Consider two sets of nodes, $U$ and $V$. $f_u(v)$ is the non-negative (integer) preference function of $u\in U$ towards $v\in V$. Similarly, $f_v(u)$ is for $v\in V$ towards $u\in U$. Let $E\subseteq U\times V$ be a set of pairs. Each pair $(u,v)\in E$ indicates that we can never allocate  one to the other.

$\pi$ is an allocation where $\pi(u)\in V$ is the node allocated to $u$ and $\pi(v)\in U$ is the node allocated to $v$. $\pi$ is weak-stable if there does not exist a pair ($u,v$) such that $f_u(v)&gt;f_u(\pi(u))$ and $f_v(u)&gt;f_v(\pi(v))$, which means that $u$ prefers $v$ to $\pi(u)$ and $v$ prefers $u$ to $\pi(v)$ simultaneously.

I want to compute a stable matching with maximum size, i.e. $maximize_{\pi}|\pi|$. Notice that, ties are allowed in any preference function.
Your problem is equivalent to MAX SMTI (Stable Marriage with Ties and Incomplete lists).  You can find the current best approximation algorithm for MAX SMTI in the following paper:
Z. Kiraly, Linear Time Local Approximation Algorithm for Maximum
Stable Marriage, Algorithms 2013, 6, 471-484.

There are many papers on MAX SMTI, but unfortunately I am not aware of any paper that handles the weighted version of MAX SMTI.