en>Kllwiki: /* History */

2014-07-23T17:28:48Z

History

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	~~The goal of the~~ '''Christofides [[approximation algorithm]]''' (named after Nicos Christofides) is to find a solution to the instances of the [[traveling salesman problem]] where the edge weights satisfy the [[triangle inequality]].		Myrtle Benny is how I'm called and I really feel comfy when people use the full name. The favorite pastime for my kids and me is to perform baseball but I haven't made a dime with it. My working day job is a meter reader. California is our beginning location.<br><br>Here is my site :: [http://www.xn--80aeerfhjjg1bv8c.xn--p1ai/%D0%A1%D0%BE%D0%B4%D0%B5%D1%80%D0%B6%D0%B0%D0%BD%D0%B8%D0%B5/cures-yeast-infection-tips-use-now home std test]
	Let <math>G=(V,w)</math> be an instance of TSP, i.e. <math>G</math> is a complete graph on the set <math>V</math> of vertices with weight function <math>w</math> assigning a nonnegative real weight to every edge of <math>G</math>.

	~~== Algorithm ==~~
	~~In [[pseudo-code]]:~~
	~~# Create a [[minimum spanning tree]] <math>T</math> of <math>G</math>.~~
	~~# Let <math>O</math> be the set of vertices with odd [[degree (graph theory)\|degree]] in <math>T</math>~~ and ~~find a [[perfect matching]] <math>M</math> with minimum weight in the [[complete graph]] over~~ the ~~vertices from <math>O</math>~~.
	~~# Combine the edges of <math>M</math>~~ and ~~<math>T</math>~~ to ~~form~~ a ~~[[multigraph]] <math>H</math>.~~
	~~# Form an [[Eulerian circuit]] in <math>H</math> (H is Eulerian because it is [[Eulerian path#Properties\|connected,~~ with ~~only even-degree vertices]]).~~
	~~# Make the circuit found in previous step [[Hamiltonian circuit\|Hamiltonian]] by skipping visited nodes (''shortcutting'').~~

	~~== Approximation ratio ==~~

	~~The cost of the solution produced by the algorithm is within 3/2 of the optimum.~~

	~~The proof is as follows:~~

	~~Let {{math\|<var>A</var>}} denote the edge set of the optimal solution of TSP for {{math\|<var>G</var>}}. Because {{math\|<var>(V,A)</var>}} is connected,~~ it ~~contains some spanning tree {{math\|<var>T</var>}} and thus {{math\|<var>w(A)</var> ≥ <var>w(T)</var>}}~~. Further let <math>B</math> denote the edge set of the optimal solution of TSP for the complete graph over vertices from <math>O</math>. Because the edge weights are triangular (so visiting more nodes cannot reduce total cost), we know that
	~~{{math\|<var>w(A)</var> ≥ <var>w(B)</var>}}. We show that there is a perfect matching of vertices from <math>O</math> with weight under~~
	~~{{math\|<var>w(B)/2</var> ≤ <var>w(A)/2</var>}} and therefore we have the same upper bound for <math>M</math> (because <math>M</math>~~ is a ~~perfect matching of minimum cost).~~
	~~Because <math>O</math> must contain an even number of vertices, a perfect matching exists. Let~~
	~~{{math\|<var>e</var><sub>1</sub>,...,<var>e</var><sub>2k</sub>}} be the (only) Eulerian path in <math>(O,B)</math>. Clearly both~~
	~~{{math\|<var>e</var><sub>1</sub>,<var>e</var><sub>3</sub>,...,<var>e</var><sub>2k-1</sub>}} and~~
	~~{{math\|<var>e</var><sub>2</sub>,<var>e</var><sub>4</sub>,..~~.~~,<var>e</var><sub>2k</sub>}} are perfect matchings and the weight of at least one of them~~ is
	~~less than or equal to {{math\|<var>w(B)/2</var>}}~~.
	~~Thus {{math\|~~<~~var~~>~~w(M)+w(T)~~<~~/var~~> ~~≤ <var>w(A) + w(A)/2</var>}} and from the triangle inequality it follows that the algorithm~~ is ~~3/2-approximative.~~

	~~== Example ==~~
	~~{\| class="wikitable"~~
	\|-
	~~\|[[File~~:~~Metrischer Graph mit 4 Knoten.svg\|200px]]\|\| Given~~: ~~metric graph <math>G=\left(V,E\right)</math> with edge weights~~
	\|-
	\|[~~[File~~:~~Christofides MST.svg\|200px]] \|\| Calculate [[minimum spanning tree]] <math>T<~~/~~math>~~.
	\|-
	~~\|[[File:V'.svg\|200px]] \|\| Calculate the set of vertices <math>V'</math> with odd degree in <math>T</math>.~~
	\|-
	~~\|[[File:G V'~~.~~svg\|200px]] \|\| Reduce <math>G</math> to the vertices of <math>V'</math> (<math>G\|_{V'}</math>).~~
	\|-
	~~\|[[File:Christofides Matching.svg\|200px]] \|\| Calculate matching <math>M</math> with minimum weight in <math>G\|_{V'}</math>.~~
	\|-
	~~\|[[File:TuM.svg\|200px]] \|\| Unite matching and spanning tree (<math>T\cup M<~~/~~math>).~~
	\|-
	~~\|[[File:Eulertour.svg\|200px]] \|\| Calculate Euler tour on <math>T\cup M<~~/~~math> (A-B-C-A-D-E-A).~~
	\|-
	~~\|[[File:Eulertour_bereinigt.svg\|200px]] \|\| Remove reoccuring vertices and replace by direct connections (A-B~~-C-D-E-~~A). In metric graphs, this step can not lengthen the tour.~~
	~~This tour is the algorithms output.~~
	\|}


	~~== References ==~~
	* [http://www.nist.gov/dads/HTML/christofides.html NIST Christofides Algorithm Definition]
	* Nicos Christofides, Worst-~~case analysis of a new heuristic for the travelling salesman problem, Report 388, Graduate School of Industrial Administration, CMU, 1976.~~

	~~[[Category:Travelling salesman problem]]~~
	~~[[Category:Graph algorithms]]~~
	~~[[Category:Spanning tree]]~~
	~~[[Category:Approximation algorithms]~~]

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en>Kllwiki: /* History */

en>Caltas: Reverted edits by 180.234.116.19 (talk) to last revision by Addbot (HG)

67.190.19.98: /* Element properties */