随机算法 (Spring 2014)/Problem Set 4: Difference between revisions
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Consider the following Markov chain for proper <math>q</math>-colorings of a graph <math>G(V,E)</math>: | Consider the following Markov chain for proper <math>q</math>-colorings of a graph <math>G(V,E)</math>: | ||
{{Theorem|Markov Chain for Graph Coloring| | {{Theorem|Markov Chain for Graph Coloring| | ||
:Start with a proper coloring of <math>G(V,E)</math>. At each step: | :Start with a proper <math>q</math>-coloring of <math>G(V,E)</math>. At each step: | ||
# Pick a vertex <math>v\in V</math> and a color <math>c\in[q]</math> uniformly at random. | # Pick a vertex <math>v\in V</math> and a color <math>c\in[q]</math> uniformly at random. | ||
# Change the color of <math>v</math> to <math>c</math> if the resulting coloring is proper; do nothing if otherwise. | # Change the color of <math>v</math> to <math>c</math> if the resulting coloring is proper; do nothing if otherwise. |
Revision as of 07:38, 2 June 2014
Problem 1
A proper [math]\displaystyle{ q }[/math]-coloring of a graph [math]\displaystyle{ G(V,E) }[/math] is a mapping [math]\displaystyle{ f:V\to [q] }[/math] such that for any edge [math]\displaystyle{ uv\in E }[/math] we have [math]\displaystyle{ f(u)\neq f(v) }[/math].
Consider the following Markov chain for proper [math]\displaystyle{ q }[/math]-colorings of a graph [math]\displaystyle{ G(V,E) }[/math]:
Markov Chain for Graph Coloring - Start with a proper [math]\displaystyle{ q }[/math]-coloring of [math]\displaystyle{ G(V,E) }[/math]. At each step:
- Pick a vertex [math]\displaystyle{ v\in V }[/math] and a color [math]\displaystyle{ c\in[q] }[/math] uniformly at random.
- Change the color of [math]\displaystyle{ v }[/math] to [math]\displaystyle{ c }[/math] if the resulting coloring is proper; do nothing if otherwise.
Show that the Markov chain is:
- aperiodic;
- irreducible if [math]\displaystyle{ q\ge \Delta+2 }[/math];
- with uniform stationary distribution.