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| The '''multi-commodity flow problem''' is a [[flow network|network flow]] problem with multiple commodities (flow demands) between different source and sink nodes.
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| ==Definition==
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| Given a flow network <math>\,G(V,E)</math>, where edge <math>(u,v) \in E</math> has capacity <math>\,c(u,v)</math>. There are <math>\,k</math> commodities <math>K_1,K_2,\dots,K_k</math>, defined by <math>\,K_i=(s_i,t_i,d_i)</math>, where <math>\,s_i</math> and <math>\,t_i</math> is the '''source''' and '''sink''' of commodity <math>\,i</math>, and <math>\,d_i</math> is the demand. The flow of commodity <math>\,i</math> along edge <math>\,(u,v)</math> is <math>\,f_i(u,v)</math>. Find an assignment of flow which satisfies the constraints:
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| :{|
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| | '''Capacity constraints''': || <math>\,\sum_{i=1}^{k} f_i(u,v) \leq c(u,v)</math>
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| | '''Flow conservation''': || <math>\,\sum_{w \in V} f_i(u,w) = 0 \quad \mathrm{when} \quad u \neq s_i, t_i </math>
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| |||<math>\,\forall v, u,\, f_i(u,v) = -f_i(v,u)</math>
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| | '''Demand satisfaction''': || <math>\,\sum_{w \in V} f_i(s_i,w) = \sum_{w \in V} f_i(w,t_i) = d_i</math>
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| |}
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| In the '''minimum cost multi-commodity flow problem''', there is a cost <math>a(u,v) \cdot f(u,v)</math> for sending flow on <math>\,(u,v)</math>. You then need to minimize
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| :<math>\sum_{(u,v) \in E} \left( a(u,v) \sum_{i=1}^{k} f_i(u,v) \right)</math>
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| In the '''maximum multi-commodity flow problem''', there are no hard demands on each commodity, but the total throughput is maximised:
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| :<math>\sum_{i=1}^{k} \sum_{w \in V} f_i(s_i,w)</math>
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| In the '''maximum concurrent flow problem''', the task is to maximise the minimal fraction of the flow of each commodity to its demand:
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| :<math>\min_{1 \leq i \leq k} \frac{\sum_{w \in V} f_i(s_i,w)}{d_i}</math>
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| ==Relation to other problems==
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| The minimum cost variant is a generalisation of the [[minimum cost flow problem]]. Variants of the [[circulation problem]] are generalisations of all flow problems.
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| ==Usage==
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| [[Routing and wavelength assignment]] (RWA) in [[optical burst switching]] of [[SONET|Optical Network]] would be approached via multi-commodity flow formulas.
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| ==Solutions==
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| In the decision version of problems, the problem of producing an integer flow satisfying all demands is [[NP-complete]],<ref name="EIS76">{{cite journal | author = S. Even and A. Itai and A. Shamir | title = On the Complexity of Timetable and Multicommodity Flow Problems | publisher = SIAM | year = 1976 | journal = SIAM Journal on Computing | volume = 5 | pages = 691–703 | url = http://link.aip.org/link/?SMJ/5/691/1 | doi = 10.1137/0205048 | issue = 4}}</ref> even for only two commodities and unit capacities (making the problem [[strongly NP-complete]] in this case).
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| If fractional flows are allowed, the problem can be solved in polynomial time through [[linear programming]].<ref>{{cite book | author = [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]] | title = [[Introduction to Algorithms]] | origyear = 1990 | edition = 2nd | year = 2001 | publisher = MIT Press and McGraw–Hill | pages = 788–789 | chapter = 29 | isbn = 0-262-03293-7}}</ref> Or through (typically much faster) [[fully polynomial time approximation scheme]]s.<ref name="">{{cite conference | author = George Karakostas | title = Faster approximation schemes for fractional multicommodity flow problems | booktitle = Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms | year = 2002 | isbn = 0-89871-513-X | pages = 166–173}}</ref>
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| ==External resources==
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| * Papers by Clifford Stein about this problem: http://www.columbia.edu/~cs2035/papers/#mcf
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| * Software solving the problem: http://typo.zib.de/opt-long_projects/Software/Mcf/
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Multi-Commodity Flow Problem}}
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| [[Category:Network flow]]
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