Improved Bounds for Perfect Sampling of - PowerPoint Presentation

Slide1

Improved Bounds for Perfect Sampling of - Colorings in Graphs

 

Joint work with Siddharth Bhandari*

Sayantan Chakraborty*

*TIFR, Mumbai

Slide2

Problem Statement and ResultGiven: graph

;

max degree

;

set of colors

Set of proper colorings

;

uniform dist over

 

Task: sample efficiently exactly according to In general, NP hard to determine if ; we assume Notice that naive accept-reject doesn’t work as

 

Previous Work: (Huber’98)

 

Current Work:

 

Goal: min value of st can efficiently sampling from with

 

Slide3

Glauber Dynamics (GD) for

Colorings

Stationary

dist

: uniform over

 

Choose

uar Choose

Update

 

 

 

 

 

Markov Chain on state space

 

 

 

 

Slide4

Approximate Sampling of Colorings

Weaker Demand:

efficiently sample

st

;

is the error parameter

 

Start with fixed

apply (GD) times  Typically coupling shows where

GD is said to be ‘fast-mixing’

 However, we need above strategy is

infeasible!

 

 

 

 

 

 

 

 

 

 

 

 

Slide5

Approximate Sampling vs Perfect Sampling of Colorings

Bubley

& Dyer’97:

 

Perfect

Sampling

Approximate Sampling

Jerrum’95:

 Vigoda’00:  Chen, Delcourt, Moitra, Perarnau & Postle’19:

 

Huber’98:

 

Feng, Guo and Yin’19:

 

Liu, Sinclair & Srivastava’19:

;

 

Current work

:

Breaks quadratic barrier for general graphs; comparable to appx

sampling

 

Slide6

Framework for Perfect Sampling

Given a MC

with stationary

dist

can we sample exactly from

???

 Beautifully answered by Propp and Wilson’96; Coupling from the Past (CFTP)Generate seq of random updates

is a random function where  Find first index st

; Output

Claim:

!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Slide7

Toy Example: CFTP for Random Walk on

 

Usual Random Walk on

stationary

dist

Uniform on

 

For each and One step of RW @   is described by

uar and independent across time

 

CFTP for RWFind first index

st

;

 

Output

:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

if

 

if

 

Slide8

Comments about CFTP

 

 

 

 

 

trajectory from

to

looks like:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

can be thought as joint evolution of various MCs: one for each

 

all MCs have coalesced by

 

Slide9

Comments about CFTP…

contd

Intuition for correctness of CFTP:

ideally start @ fixed

and iterate forever:

 

 

 

     

Ergodicity of

 

To implement above: shift all indices by

 

 

 

 

 

 

 

 

 

If

then the value of

doesn’t matter!!!

 

Efficiency

: need

to be small; this depends on

and

itself

 

Slide10

CFTP for Colorings

Underlying MC is Glauber Dynamics;

set of all

colorings of

 

Input: graph

with

and max degree

 

is described by

and

uar and independent across time;

 

for all

;

first color outside

acc to

 

Clearly

is GD

 

Issues:

Time:

how long before

?

Space:

how to keep track?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Slide11

Return of the Toy Example

Possible to check

by keeping tack of states

If

then

 

No natural order for colorings!

 

 

 

 

 

 

 

 

 

 

Slide12

Bounding Chain (BC)

Generalization of previous idea: Huber’98 and

Haggstrom

& Nelander’99

Keeps track of exp colorings efficiently

BC is a MC on

at time

at vertex :

is a list of colors

 Suppose we want to check ?; BC can check efficiently wo false positives!BC also shows: if large & then

 

Slide13

Bounding Chain…

contd

BC is a MC on

at time

at vertex

:

is a list of colors

 

Start:

 Invariant: at time at vertex :

In other words:

upper bounds possible colors at

at time

;

a poly-sized object can contain exp many colorings

 

End:

If

for all

then

 

 

 

 

 

Slide14

Bounding Chain a la Huber

is described by

and

uar and independent across time;

 

for all

;

first color outside

acc to

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Recap CFTP

:

set of all

colorings of

 

Slide15

Bounding Chain a la Huber…

contd

 

 

 

 

 

 

 

 

 

 

 

Warm-up Phase

 

 

 

 

Warm-up: O(

)

 

Coalescence: O(

)

 

 

 

Slide16

Bounding Chain a la Huber…Warm-Up Phase

 

Warm-up Phase

 

 

 

Warm-up: O(

)

 

Coalescence: O(

)

 

 

 

Invariant:

clear at

as

for all

inductively if

then

 

 

 

 

 

 

End of Warm-up Phase @

whp

 

Slide17

Bounding Chain a la Huber…Coalescence Phase

 

Coalescence: O(

)

 

 

 

 

End of Warm-up Phase @

whp

 

Definition

:

Notice that for any

then

is superset of blocked colors

 

 

 

 

 

 

 

 

Case (a) is progress; large

is fav!

Case (b) is regress; small

is

unfav

!

 

 

 

Slide18

Bounding Chain a la Huber…Conclusion

Definition

:

want

all list-sizes are

 

performs RW on

reflecting &

absorbing  Huber showed that there is +ve () drift if if

then

 

In fact, for +

ve

drift need:

 

Slide19

New Bounding Chain

 

Warm-up: O(

)

 

Coalescence: O(

)

 

End of Warm-up @ :

with prob

 Recall

;

want

 Coalescence phase

performs RW on

reflecting &

absorbing

 

Max list size during coal phase:

 

For +

ve

drift need:

 

Main idea

: changing underlying CFTP process while respecting GD

st

uncertainty of colors (list size) at any vertex is smaller

can be of types:

Compress

or Contract

 

Slide20

Let

 

:

 

Toss a

-biased coin :

If HEADS: use

If TAILS: use if possible () o/w use

  exists if: (equating mass on ) Needs ; we ensure this by requiring  Why not set

?

Not enough mass on ! 

 

 

 

 

 

 

Fix : Sample

uar

and let

 

 

 

Contract Updates :

Biased

Sampling

is a restricted set.

Want

:

Sample

uar

from

 

Slide21

Contract

Updates : Biased Sampling

 

 

 

 

Contract

 

Contract

is specified by

 

 

 

Slide22

Contract Updates : Biased Sampling chooses

wp

 

Always produce lists of size 2

Once all list sizes go down to 2 they stay that way for

 

Issue :

Never produce lists of siz

e 1 !Fix : Underestimate the probability with which chooses  Issue : does not have access to the value of  Fix : underestimates by tossing a coin

Monotonically couple

and  

Slide23

Contract

Updates : Biased Sampling

 

 

 

 

Contract

 

 

Contract

is specified by

 

 

 

Slide24

Compress Updates : Biased Sampling Again

 

Updating

u

Pick

uniformly at random

 

 

Compress

 

 

Compress

is specified by

 

 

 

Slide25

Compress Updates : Biased Sampling Again

 

Updating

Pick

uniformly at random

If

choose nothing

Case I :

 

Set  

 

 

 

 

Compress

 

Slide26

 

 

 

 

Compress

 

Case II :

 

Flip a coin

w.p

. of heads

If heads,

Else

 

 

Compress Updates : Biased Sampling Again

Slide27

New Bounding

Chain : Warm-Up Phase

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CFTP

BC

Compress

Compress

Compress

Contract

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Contract

 

Compress

 

Compress

 

Compress

 

Focus on bringing list at

to

size

 

Slide28

New BC…Coalescence Phase

 

Warm-up: O(

)

 

Coalescence: O(

)

 

End of Warm-up @ :

with prob

 Recall

;

want

 Coalescence phase

performs RW on

reflecting &

absorbing

 

Max list size during coal phase:

 

For +

ve

drift need:

 

Slide29

New BC…Coalescence Phase…

contd

 

 

 

 

 

 

 

 

 

 

CFTP

Contract

Contract

Contract

Contract

generated

iid

across time;

evolves accordingly

 

With

else

 

Slide30

Open Question

Can we push the bound to

is the natural limit for coloring which comes out path coupling

At least, a Grand Coupling exists at

 

Thank you

Slide31

Contract Update

 

 

 

 

Contract

 

Contract

is specified by

 

 

 

End of Spruce-up @

:

 

 

 

 

 

Contract

 

 

End of Contract @

+1

:

Also, w prob

 

Slide32

Compress Update

 

 

 

 

 

 

 

 

 

 

CFTP

Compress

Compress

Compress

Contract

 

 

 

 

 

Compress

is specified by

 

 

 

 

 

 

 

Compress

 

Compress