|
|
| Line 1: |
Line 1: |
| 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]]
| |
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.
Here is my site :: home std test