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A different view of independent sets in bipartite graphs. Qi Ge Daniel Štefankovič University of Rochester. A different view of independent sets in bipartite graphs. counting/sampling independent sets in general graphs:. polynomial time sampler for

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slide1

A different view of independent sets in

bipartite graphs

Qi Ge

Daniel Štefankovič

University of Rochester

slide2

A different view of independent sets in

bipartite graphs

counting/sampling independent sets in general graphs:

polynomial time sampler for

 5 (Dyer,Greenhill ’00, Luby,Vigoda’99, Weitz’06).

no polynomial time sampler (unless NP=RP) for

  25 (Dyer, Frieze, Jerrum ’02).

Glauber dynamics does not mix in polynomial time

for 6-regular bipartite graphs (example: union of 6

random matchings) (Dyer, Frieze, Jerrum ’02).

 = maximum degree of G

slide3

A different view of independent sets in

bipartite graphs

counting/sampling independent sets in bipartite graphs:

polynomial time sampler for

 5 (Dyer,Greenhill ’00, Luby,Vigoda’99, Weitz’06).

no polynomial time sampler (unless NP=RP) for

  25 (Dyer, Frieze, Jerrum ’02).

(max idependent set in bipartite graph  max matching)

Glauber dynamics does not mix in polynomial time

for 6-regular bipartite graphs (example: union of 6

random matchings) (Dyer, Frieze, Jerrum ’02).

 = maximum degree of G

slide4

How hard is counting/sampling

independent sets in bipartite graphs?

Why do we care?

* bipartite independent sets

equivalent to

* enumerating solutions of a linear Datalog program

* downsets in a poset

(Dyer, Goldberg, Greenhill, Jerrum’03)

* ferromagnetic Ising with mixed external field

(Goldberg,Jerrum’07)

* stable matchings

(Chebolu, Goldberg, Martin’10)

slide5

A different view of independent sets in

bipartite graphs

0 0

0 0

0 0

0 1

1 0

1 0

1 0

1 1

0 1

0 0

0 0

1 0

1 1

1 1

0 1

1 1

1 0

0 0

0 0

1 1

0 1

1 0

1 1

0 1

Ge, Štefankovič ’09

1 1

0 0

0 1

0 1

1 0

0 1

1 1

1 0

Independent sets in

a bipartite graph.

0-1 matrices weighted by

(1/2)rank (1 allowed at Auv

if uv is an edge)

slide6

A different view of independent sets in

bipartite graphs

0 0

0 0

1 0

1 0

0 1

0 0

0 0

1 0

1 0

0 0

0 1

1 0

Ge, Štefankovič ’09

1 1

0 0

1 1

1 0

Independent sets in

a bipartite graph.

0-1 matrices weighted by

(1/2)rank (1 allowed at Auv

if uv is an edge)

#IS = 2|VU| - |E| 2-rk(A)

A  B

slide7

A different view of independent sets in

bipartite graphs

Question:

Is there a polynomial-time sampler

that produces matrices A  B with

P(A)  2-rank(A)

0 0

0 0

1 0

1 0

0 1

0 0

0 0

1 0

1 0

0 0

0 1

1 0

Ge, Štefankovič ’09

1 1

0 0

1 1

1 0

Bij=0  Aij=0

Independent sets in

a bipartite graph.

0-1 matrices weighted by

(1/2)rank (1 allowed at Auv

if uv is an edge)

#IS = 2|V U| - |E| 2-rk(A)

(everything over the F2)

A  B

slide8

Natural MC

flip random entry +

Metropolis filter.

A = Xt with random (valid)

entry flipped

if rank(A)  rank(Xt)

then Xt+1 = A

if rank(A) > rank(Xt) then

Xt+1 = A w.p. ½

Xt+1 = Xt w.p. ½

we conjectured it is mixing

BAD NEWS:

Goldberg,Jerrum’10: the chain is exponentially

slow for some graphs.

slide9

Our inspiration (Ising model):

Fortuin-Kasteleyn

Ising model: assignment of spins

to sites weighted by the number

of neighbors that agree

Random cluster model: subgraphs

weighted by the number of

components and the number of

edges

Newell Montroll ‘53

High temperature expansion:

even subgraphs weighted

by the number of edges

slide10

Random cluster model

Z(G,q,)=  q(S)|S|

SE

number of connected

components of (G,S)

(Tutte polynomial)

Ising model

Potts model

chromatic polynomial

Flow polynomial

slide11

Random cluster model

R2 model

R2(G,q,)=  qrk(S)|S|

Z(G,q,)=  q(S)|S|

2

SE

SE

number of connected

components of (G,S)

rank (over F2) of the

adjacency matrix of (G,S)

Matchings

Perfect matchings

Independent sets

(for bipartite only!)

(Tutte polynomial)

Ising model

Potts model

chromatic polynomial

Flow polynomial

More ?

slide12

Complexity of exact evaluation

R2(G,q,)=  qrk (S)|S|

2

SE

Tutte polynomial

R2 model

spanning trees

BIS

2|E|-|V|+|isolated V|

q

Ge, Štefankovič ’09

Jaeger, Vertigan, Welsh ’90

easy if (x-1)(y-1)=1, or

(1,1),(-1,-1),(0,-1),(-1,0)

easy if q{0,1}

or =0, or (1/2,-1)

#P-hard elsewhere

#P-hard elsewhere (GRH)

slide13

“high-temperature expansion”

(1-((u),(v))

2|E| #BIS =

U{0,1}

V{0,1}

{u,v}E

where

(1,1) = 1

(0,1) = (1,0) = (0,0) = -1

slide14

“high-temperature expansion”

(1-((u),(v))

2|E| #BIS =

U{0,1}

V{0,1}

{u,v}E

where

(1,1) = 1

(0,1) = (1,0) = (0,0) = -1

(-1)|S| ((u),(v))

=

U{0,1}

SE

V{0,1}

{u,v}S

slide15

“high-temperature expansion”

(1-((u),(v))

2|E| #BIS =

U{0,1}

V{0,1}

{u,v}E

where

(1,1) = 1

(0,1) = (1,0) = (0,0) = -1

(-1)|S| ((u),(v))

=

U{0,1}

SE

V{0,1}

{u,v}S

0 if some v V has an odd number of neighbors

in (UV,S) labeled by 1

(-2)|V| otherwise

= {

slide16

“high-temperature expansion”

2|E| #BIS =

(-1)|S| ((u),(v))

=

U{0,1}

SE

V{0,1}

{u,v}S

2|V|

=

number of u such that uTA = 0 (mod 2)

SE

bipartite adjacency matrix of (UV,S)

2|V|+|U| 

2- rank (A))

=

2

SE

slide17

“high-temperature expansion” – curious

f(A,) =  |v| ( )|Av|

1-

1

1

1+

f(A,1) = 2rank (A)

2

T

f(A,1) = f(A,1)

But in fact:

T

f(A,) = f(A,)

slide18

Questions:

Is there a polynomial-time sampler that produces

matrices A  B with P(A)  2-rank(A) ?

What other quantities does the R2 polynomial

encode ?

R2(G,q,)=  qrk(S)|S|

2

SE