The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. The minimum cost flow problem is one of the most fundamental among all flow and circulation problems because most other such problems can be cast as a minimum cost flow problem and also that it can be solved efficiently using the network simplex algorithm.

A flow network is a directed graph ${\displaystyle G=(V,E)}$ with a source vertex ${\displaystyle s\in V}$ and a sink vertex ${\displaystyle t\in V}$, where each edge ${\displaystyle (u,v)\in E}$ has capacity ${\displaystyle c(u,v)>0}$, flow ${\displaystyle f(u,v)}$ and cost ${\displaystyle a(u,v)}$, with most minimum-cost flow algorithms supporting edges with negative costs. The cost of sending this flow along an edge ${\displaystyle (u,v)}$ is ${\displaystyle f(u,v)\cdot a(u,v)}$. The problem requires an amount of flow ${\displaystyle d}$ to be sent from source ${\displaystyle s}$ to sink ${\displaystyle t}$.

The definition of the problem is to minimize the total cost of the flow over all edges:

with the constraints

A variation of this problem is to find a flow which is maximum, but has the lowest cost among the maximum flow solutions. This could be called a minimum-cost maximum-flow problem and is useful for finding minimum cost maximum matchings.

With some solutions, finding the minimum cost maximum flow instead is straightforward. If not, one can find the maximum flow by performing a binary search on ${\displaystyle d}$.

A related problem is the minimum cost circulation problem, which can be used for solving minimum cost flow. The minimum cost circulation problem has no source and sink; instead it has costs and lower and upper bounds on each edge, and seeks flow amounts within the given bounds that balance the flow at each vertex and minimize the sum over edges of cost times flow. Any minimum-cost flow instance can be converted into a minimum cost circulation instance by setting the lower bound on all edges to zero, and then making an extra edge from the sink ${\displaystyle t}$ to the source ${\displaystyle s}$, with capacity ${\displaystyle c(t,s)=d}$ and lower bound ${\displaystyle l(t,s)=d}$, forcing the total flow from ${\displaystyle s}$ to ${\displaystyle t}$ to also be ${\displaystyle d}$.

The following problems are special cases of the minimum cost flow problem (we provide brief sketches of each applicable reduction, in turn):