Here is the set-up:
Let $G = (V(G), E(G))$ be a bipartite graph with $ V_1 \sqcup V_2$ denoting the vertices with respect to 2-colouring of the graph. Consider the induced directed graph $G' = (V(G),A(G))$ with the vertices $V(G)$ and with an arrow of each direction replacing each edge of $G$. Furthermore, we add a weighting $ wt : A(G) \rightarrow \mathbb{R}$ to each arrow such that $wt(u \rightarrow v)*wt(v \rightarrow u) = 1$ for all adjacent $u,v \in V(G)$.
Here is the question:
Does there exist a weighting function such that such that if $u, v \in V_i $ then
$$\sum_{x \in N(u)} wt(x\rightarrow u) = \sum_{x \in N(v) } wt(x\rightarrow v) $$
where $N(v)$ is the set of adjacent vertices to $v$ in $G$ and $V_i \in \{V_1,V_2\}$.
I'm very ignorant of this general area. It seems like this might be a well-known example of some discrete version of some kind of heat equation or harmonic function. Any references to anything related would be extremely useful to me! Thanks!
The constraint on the weighting is not very easy to deal with... because it's about the product of the weights, right ? Somehow it doesn't seem natural.
Anyway : sometimes it's impossible (take $V_1 = \{a,b,c\}, V_2 = \{d,e\}, E=\{(ad), (bd), (ce)\}$ : the constraints on $V_1$ say that all weights are equal, and the constraints on $V_2$ say that they are not.), sometimes it will be easy (you have $|V|-2+|E|$ constraints and $2|E|$ variables) so you should have enough degrees of freedom in most instances of the problem).
You could transform this into an instance of semidefinite programming (your constraints are quadratic, defined with a dot product) so this can be solved "efficiently", for example with interior point methods.
Maybe there is a clever way to reformulate this into a classical problem, but I can't seem to find an idea that could make this quadratic constraint appear magically.