Multi-commodity flow problem

The multi-commodity flow problem is a network flow problem with multiple commodities (flow demands) between different source and sink nodes.

Definition
Given a flow network $$\,G(V,E)$$, where edge $$(u,v) \in E$$ has capacity $$\,c(u,v)$$. There are $$\,k$$ commodities $$K_1,K_2,\dots,K_k$$, defined by $$\,K_i=(s_i,t_i,d_i)$$, where $$\,s_i$$ and $$\,t_i$$ is the source and sink of commodity $$\,i$$, and $$\,d_i$$ is its demand. The variable $$\,f_i(u,v)$$ defines the fraction of flow $$\,i$$ along edge $$\,(u,v)$$, where $$\,f_i(u,v) \in [0,1]$$ in case the flow can be split among multiple paths, and $$\,f_i(u,v) \in \{0,1\}$$ otherwise (i.e. "single path routing"). Find an assignment of all flow variables which satisfies the following four constraints:

(1) Link capacity: The sum of all flows routed over a link does not exceed its capacity.
 * $$\forall (u,v)\in E:\,\sum_{i=1}^{k} f_i(u,v)\cdot d_i \leq c(u,v)$$

(2) Flow conservation on transit nodes: The amount of a flow entering an intermediate node $$u$$ is the same that exits the node.
 * $$\forall i\in\{1,\ldots,k\}:\,\sum_{w \in V} f_i(u,w) - \sum_{w \in V} f_i(w,u) = 0 \quad \mathrm{when} \quad u \neq s_i, t_i $$

(3) Flow conservation at the source: A flow must exit its source node completely.
 * $$\forall i\in\{1,\ldots,k\}:\,\sum_{w \in V} f_i(s_i,w) - \sum_{w \in V} f_i(w,s_i) = 1$$

(4) Flow conservation at the destination: A flow must enter its sink node completely.
 * $$\forall i\in\{1,\ldots,k\}: \,\sum_{w \in V} f_i(w,t_i) - \sum_{w \in V} f_i(t_i,w) = 1$$

Corresponding optimization problems
Load balancing is the attempt to route flows such that the utilization $$U(u,v)$$ of all links $$(u,v)\in E$$ is even, where


 * $$U(u,v)=\frac{\sum_{i=1}^{k} f_i(u,v)\cdot d_i}{c(u,v)}$$

The problem can be solved e.g. by minimizing $$\sum_{u,v\in V} (U(u,v))^2$$. A common linearization of this problem is the minimization of the maximum utilization $$U_{max}$$, where


 * $$\forall (u,v)\in E:\, U_{max} \geq U(u,v)$$

In the minimum cost multi-commodity flow problem, there is a cost $$a(u,v) \cdot f(u,v)$$ for sending a flow on $$\,(u,v)$$. You then need to minimize


 * $$\sum_{(u,v) \in E} \left( a(u,v) \sum_{i=1}^{k} f_i(u,v)\cdot d_i \right)$$

In the maximum multi-commodity flow problem, the demand of each commodity is not fixed, and the total throughput is maximized by maximizing the sum of all demands $$\sum_{i=1}^{k} d_i$$

Relation to other problems
The minimum cost variant of the multi-commodity flow problem is a generalization of the minimum cost flow problem (in which there is merely one source $$s$$ and one sink $$t$$). Variants of the circulation problem are generalizations of all flow problems. That is, any flow problem can be viewed as a particular circulation problem.

Usage
Routing and wavelength assignment (RWA) in optical burst switching of Optical Network would be approached via multi-commodity flow formulas.

Register allocation can be modeled as an integer minimum cost multi-commodity flow problem: Values produced by instructions are source nodes, values consumed by instructions are sink nodes and registers as well as stack slots are edges.

Solutions
In the decision version of problems, the problem of producing an integer flow satisfying all demands is NP-complete, even for only two commodities and unit capacities (making the problem strongly NP-complete in this case).

If fractional flows are allowed, the problem can be solved in polynomial time through linear programming, or through (typically much faster) fully polynomial time approximation schemes.

Applications
Multicommodity flow is applied in the overlay routing in content delivery.

External resources

 * Papers by Clifford Stein about this problem: http://www.columbia.edu/~cs2035/papers/#mcf
 * Software solving the problem: https://web.archive.org/web/20130306031532/http://typo.zib.de/opt-long_projects/Software/Mcf/