Multi-commodity flow problem

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


Given a flow network , where edge has capacity . There are commodities , defined by , where and is the source and sink of commodity , and is its demand. The variable defines the fraction of flow along edge , where in case the flow can be split among multiple paths, and 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.

(2) Flow conservation on transit nodes: The amount of a flow entering an intermediate node is the same that exits the node.

(3) Flow conservation at the source: A flow must exit its source node completely.

(4) Flow conservation at the destination: A flow must enter its sink node completely.

Corresponding optimization problems[edit]

Load balancing is the attempt to route flows such that the utilization of all links is even, where

The problem can be solved e.g. by minimizing . A common linearization of this problem is the minimization of the maximum utilization , where

In the minimum cost multi-commodity flow problem, there is a cost for sending a flow on . You then need to minimize

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

Relation to other problems[edit]

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


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


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[edit]


  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. Retrieved 2016-10-05.
  2. ^ Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2009). "29". Introduction to Algorithms (3rd ed.). MIT Press and McGraw–Hill. p. 862. ISBN 978-0-262-03384-8.CS1 maint: multiple names: authors list (link)
  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.

Add: Jean-Patrice Netter, Flow Augmenting Meshings: a primal type of approach to the maximum integer flow in a muti-commodity network, Ph.D dissertation Johns Hopkins University, 1971