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.

Definition[edit]

Given a flow network , where edge has capacity . There are commodities , defined by , where and is the source and sink of commodity , and is the demand. The flow of commodity along edge is . Find an assignment of flow which satisfies the constraints:

Capacity constraints:
Flow conservation:
Demand satisfaction:

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

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

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

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.

Usage[edit]

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

Solutions[edit]

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]

References[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. 
  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.