Improved bounds for MCMC sampling colorings of G ( n , d / n ) - PowerPoint PPT Presentation

Block Construction Weights [Efthymiou (2014)] • Each vertex u of degree deg ( u ) is assigned weight � (1 + γ ) − 1 deg ( u ) ≤ (1 + ǫ ) d W ( u ) = d c · deg ( u ) otherwise • Every path L is assigned weight � u ∈ L W ( u ) “Break Points” ln n Γ( v ) := set of paths of length at most d 2 / 5 that emanate from v . For a break-point v , we have

Block Construction Weights [Efthymiou (2014)] • Each vertex u of degree deg ( u ) is assigned weight � (1 + γ ) − 1 deg ( u ) ≤ (1 + ǫ ) d W ( u ) = d c · deg ( u ) otherwise • Every path L is assigned weight � u ∈ L W ( u ) “Break Points” ln n Γ( v ) := set of paths of length at most d 2 / 5 that emanate from v . For a break-point v , we have �� � max W ( u ) ≤ 1 . L ∈ Γ( v ) u ∈ L

How do the Blocks look like

How do the Blocks look like

How do the Blocks look like Boundary of the block Consists only of break points.

How do the Blocks look like Low degree “buffer” . . . between boundary vertices and a high degree vertex

How do the Blocks look like . . . for the analysis the effect of high degrees disappears

Proving Rapid Mixing

Proving Rapid Mixing Path Coupling, [Bubley, Dyer 1997]

Proving Rapid Mixing Path Coupling, [Bubley, Dyer 1997] Consider ( X t ), ( Y t ) such that X 0 ⊕ Y 0 = { w ∗ }

Proving Rapid Mixing Path Coupling, [Bubley, Dyer 1997] Consider ( X t ), ( Y t ) such that X 0 ⊕ Y 0 = { w ∗ } For rapid mixing it suffices to have a coupling such that E [ dist ( X 1 , Y 1 ) | X 0 , Y 0 ] ≤ (1 − γ ) dist ( X 0 , Y 0 ) ,

Proving Rapid Mixing Path Coupling, [Bubley, Dyer 1997] Consider ( X t ), ( Y t ) such that X 0 ⊕ Y 0 = { w ∗ } For rapid mixing it suffices to have a coupling such that E [ dist ( X 1 , Y 1 ) | X 0 , Y 0 ] ≤ (1 − γ ) dist ( X 0 , Y 0 ) , where � dist ( σ, τ ) = β ( u ) u ∈ σ ⊕ τ

Distance between σ and τ

Distance between σ and τ dist ( σ, τ ) depends on the block partition B .

Distance between σ and τ dist ( σ, τ ) depends on the block partition B .

Distance between σ and τ dist ( σ, τ ) depends on the block partition B .

Distance between σ and τ A distance that counts the disagreeing edges between the blocks

Distance between σ and τ A new distance metric Given G ( n , d / n ) and set of blocks B , for any two σ, τ � dist ( σ, τ ) = 1 { v ∈ σ ⊕ τ } deg out ( v ) v ∈ ∂ B

Distance between σ and τ A new distance metric Given G ( n , d / n ) and set of blocks B , for any two σ, τ dist ( σ, τ ) = n 2 � � 1 { v ∈ σ ⊕ τ } deg out ( v ) + 1 { v ∈ σ ⊕ τ } v ∈ ∂ B v ∈ V \ ∂ B

Distance between σ and τ A new distance metric Given G ( n , d / n ) and set of blocks B , for any two σ, τ dist ( σ, τ ) = n 2 � � 1 { v ∈ σ ⊕ τ } deg out ( v )+ 1 { v ∈ σ ⊕ τ } v ∈ ∂ B v ∈ V \ ∂ B

The coupling

The coupling B 2 B 0 B 3 B 1 B 4

The coupling B 2 B 0 B 3 B 1 B 4

The coupling B 2 B 0 B 3 B 1 B 4

The coupling B 2 B 0 B 3 B 1 B 4

The coupling B 2 B 0 B 3 B 1 B 4

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The coupling of X ( B ) and Y ( B )

The coupling of X ( B ) and Y ( B ) • one vertex at a time

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement • disagreement probability 1 � deg ( v ) < k k − deg ( v ) ̺ v = 1 otherwise

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement • disagreement probability 1 � deg ( v ) < k k − deg ( v ) ̺ v = 1 otherwise • probability of the most likely color

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement • disagreement probability 1 � deg ( v ) < k k − deg ( v ) ̺ v = 1 otherwise • probability of the most likely color

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement • disagreement probability 1 � deg ( v ) < k k − deg ( v ) ̺ v = 1 otherwise • probability of the most likely color

The coupling of X ( B ) and Y ( B ) • one vertex at a time • pick a vertex next to a disagreement • disagreement probability 1 � deg ( v ) < k k − deg ( v ) ̺ v = 1 otherwise • probability of the most likely color

Rapid Mixing for k > 2 d