# Max-flow problem with additional constraint

Consider the max-flow problem with a set of additional constraints, each in the following form: the flow on edge $$e$$ must equal the flow on edge $$e'$$. My question is how to modify existng max-flow algo to solve this variant.

• It is very unlikely that you can modify normal graph theoretic max-flow algorithms for your variant. For example, even when the capacities are integer valued, the only feasible max-flow could have fractional value. This implies that for example a variation of Ford-Fulkerson would not work. Nov 2, 2022 at 6:54

Given a directed graph $$G=(V,E)$$, a source $$s \in V$$ with no incoming edges, a sink $$t \in V$$ with no outgoing edges, and a non-negative capacity $$c_e$$ for each edge $$e \in E$$, the formulation has one variable $$f_e \in \mathbb{R}$$ for each edge $$e \in E$$ and it is as follows: $$\max \sum_{(s,v) \in E}f_{(s,v)} \quad \mbox{s.t.} \\ 0 \le f_e \le c_e \quad \forall e\in E \\ \sum_{(u,v) \in E} f_{(u,v)} = \sum_{(v,u) \in E} f_{(u,v)} \quad \forall v \in V \setminus \{s,t\}$$ You can directly add your constraints to this formulation in the form $$f_e = f_{e'}$$.