Consider a graph $G=(V, A, w)$, where each arc $(u,v)\in A$ has a non negative weight $w_{u,v} \in \mathbb{R}^+$, partition $V$ into $U$ and $W$, $W=V-U$ such that $\sum_{(i,j)\in A} w_{i,j}z_{i,j}$ where $z_{i,j} = [[i\in U \land j \in W]]$.
By getting a randomized cut we have a 1/4 approximation:
\begin{align} \mathbb{E}\left[\sum_{(i,j)\in A} w_{i,j} z_{i,j}\right] = \sum_{(i,j)\in A}w_{i,j}\mathbb{E}\left[z_{i,j}\right] = \sum_{(i,j)\in A}w_{i,j} \Pr(i\in U \text{ and } j\in W) \geq \frac{1}{4} OPT \end{align}
We can instead solve an integer program to get an exact solution. Consider a randomized algorithm that solves the linear relaxation of the integer program below and puts each vertex $i$ in $U$ with probability $1/4+x_i/2$.
\begin{align} \text{maximize } &\sum_{(i,j)\in A} w_{i,j} z_{i,j}\\ \text{subject to }& z_{i,j} \leq x_{i} &\ \forall (i,j) \in A\\ & z_{i,j} \leq 1- x_{j} &\forall (i,j) \in A\\ & x_{i}\in\left\{0,1\right\} &\forall i \in V\\ & 0\leq z_{i,j} \leq 1 &\forall (i,j) \in A\\ \end{align}
We can relax the integer program to a linear program by relaxing $ x_{i}\in\left\{0,1\right\}$ to $ x_{i}\in[0,1]$ for all $ i \in V$.
\begin{align} \mathbb{E}\left[\sum_{(i,j)\in A} w_{i,j} z_{i,j}\right] = \sum_{(i,j)\in A}w_{i,j}\mathbb{E}\left[z_{i,j}\right] = \sum_{(i,j)\in A}w_{i,j} \Pr(i\in U \text{ and } j\in W) = \sum_{(i,j)\in A}w_{i,j}(\frac{1}{4}+\frac{x_i}{2})\left[1-(\frac{1}{4}+\frac{x_j}{2})\right] \end{align}
I have trouble however going further and showing that this is a 1/2 approximation though.
