I'm trying to find an algorithm for the following problem. Let $G$ be a bipartite graph. The edges in $G$ have labels $R$; each label $R(u, v)$ is an integer range $[a, b]$ with $a$ and $b$ being nonnegative integers with $a \le b$. The problem is to assign each node in the graph with nonnegative integer labels $L$ such that
- If $(u, v) \in G$ then $L(v) - L(u) \in R(u, v)$
- The sum of $L(v) - L(u)$ for all $(u, v) \in G$ is minimized
Are there any similar problems that can read about to give me some insight?