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 the demand. The flow of commodity $\,i$ along edge $\,(u,v)$ is $\,f_i(u,v)$. Find an assignment of flow which satisfies the constraints:

Capacity constraints:	$\,\sum_{i=1}^{k} f_i(u,v) \leq c(u,v)$
Flow conservation:	$\,\sum_{w \in V} f_i(u,w) = 0 \quad \mathrm{when} \quad u \neq s_i, t_i$
	$\,\forall v, u,\, f_i(u,v) = -f_i(v,u)$
Demand satisfaction:	$\,\sum_{w \in V} f_i(s_i,w) = \sum_{w \in V} f_i(w,t_i) = d_i$

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

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

In the maximum multi-commodity flow problem, there are no hard demands on each commodity, but the total throughput is maximised:

$\sum_{i=1}^{k} \sum_{w \in V} f_i(s_i,w)$

In the maximum concurrent flow problem, the task is to maximise the minimal fraction of the flow of each commodity to its demand:

$\min_{1 \leq i \leq k} \frac{\sum_{w \in V} f_i(s_i,w)}{d_i}$

Relation to other problems

The minimum cost variant is a generalisation of the minimum cost flow problem. Variants of the circulation problem are generalisations of all flow problems.

Usage

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

Solutions

In the decision version of problems, the problem of producing an integer flow satisfying all demands is NP-complete,^[1] 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.^[2] Or through (typically much faster) fully polynomial time approximation schemes.^[3]

External resources

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

References

1. S. Even and A. Itai and A. Shamir (1976). "On the Complexity of Timetable and Multicommodity Flow Problems". SIAM Journal on Computing (SIAM) 5 (4): 691–703. doi:10.1137/0205048.
2. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001) [1990]. "29". Introduction to Algorithms (2nd ed.). MIT Press and McGraw–Hill. pp. 788–789. ISBN 0-262-03293-7.
3. George Karakostas (2002). "Faster approximation schemes for fractional multicommodity flow problems". Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms. pp. 166–173. ISBN 0-89871-513-X.