Preferential concentration - Revision history

en>Yobot: Tagging using AWB (10703)

2015-01-07T13:21:04Z

Tagging using AWB (10703)

← Older revision		Revision as of 15:21, 7 January 2015
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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~~.		Hello. Allow me introduce the writer. Her name is Refugia Shryock. South Dakota is where me and my husband reside and my family members enjoys it. For years he's been working as a receptionist. To gather cash is 1 of the things I adore most.<br><br>My web-site; at home std test ([http://rivoli.enaiponline.com/user/view.php?id=438251&course=1 click the following article])

	~~==Definition==~~

	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:~~

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

	~~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~~

	~~:<math>\sum_{(u,v) \in E} \left( a(u,v) \sum_{i=1}^{k} f_i(u,v) \right)</math>~~

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

	~~:<math>\sum_{i=~~1~~}^{k} \sum_{w \in V} f_i(s_i,w)</math>~~

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

	:<~~math~~>~~\min_{1 \leq i \leq k} \frac{\sum_{w \in V} f_i(s_i,w)}{d_i}~~<~~/math~~>

	~~==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 [[SONET\|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]],<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~~).

	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>

	~~==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==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Multi-Commodity Flow Problem}}~~
	~~[[Category:Network flow]]~~

en>DoctorKubla: Replacing deprecated {{wikify}} with {{underlinked}}

2012-12-13T10:24:33Z

Replacing deprecated {{wikify}} with {{underlinked}}

← Older revision		Revision as of 12:24, 13 December 2012
Line 1:		Line 1:
	~~Nice to satisfy you~~, ~~my title~~ is ~~Refugia~~. ~~One~~ of the ~~very very best things~~ in the ~~world~~ for ~~him is~~ to ~~gather badges~~ but he is ~~struggling~~ to ~~find time~~ for it. ~~Her family members lives~~ in ~~Minnesota~~. ~~Since she was 18 she'~~s ~~been operating as a receptionist but her marketing by no means arrives~~.<br><br>~~Stop~~ by ~~my webpage - [~~http://www.~~streaming~~.~~iwarrior.net~~/~~blog~~/~~258268~~ http://~~www~~.~~streaming~~.~~iwarrior.net~~/]		The '''multi-commodity flow problem''' is a [[flow network\|network flow]] problem with multiple commodities (flow demands) between different source and sink nodes.

			==Definition==

			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:

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

			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

			:<math>\sum_{(u,v) \in E} \left( a(u,v) \sum_{i=1}^{k} f_i(u,v) \right)</math>

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

			:<math>\sum_{i=1}^{k} \sum_{w \in V} f_i(s_i,w)</math>

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

			:<math>\min_{1 \leq i \leq k} \frac{\sum_{w \in V} f_i(s_i,w)}{d_i}</math>

			==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 [[SONET\|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]],<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).

			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>

			==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==
			<references/>

			{{DEFAULTSORT:Multi-Commodity Flow Problem}}
			[[Category:Network flow]]

en>BattyBot: merged templates into Template:Multiple issues & general fixes, added wikify tag using AWB (7961)

2012-03-04T02:08:01Z

merged templates into Template:Multiple issues & general fixes, added wikify tag using AWB (7961)

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