A different view of independent sets in bipartite graphs

Qi GeDaniel Štefankovič

University of Rochester

counting/sampling independent sets in general graphs:

A different view of 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).

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

counting/sampling independent sets in bipartite graphs:

A different view of 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).

Glauber dynamics does not mix in polynomial time for 6-regular bipartite graphs (example: union of 6 random matchings) (Dyer, Frieze, Jerrum ’02).

(max idependent set in bipartite graph max matching)

= maximum degree of G

Why do we care?

How hard is counting/sampling independent sets in bipartite graphs?

* 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)

0 00 0

0 00 1

1 01 0

1 01 1

0 10 0

0 01 0

1 11 1

0 11 1

1 00 0

0 01 1

0 11 0

1 10 1

1 10 0

0 10 1

1 00 1

1 11 0

Independent sets in a bipartite graph. 0-1 matrices weighted by

(1/2)rank (1 allowed at Auv if uv is an edge)

Ge, Štefankovič ’09

A different view of independent sets in bipartite graphs

0 00 0

1 01 0

0 10 0

0 01 0

1 00 0

0 11 0

1 10 0

1 11 0

A different view of independent sets in bipartite graphs

Ge, Štefankovič ’09

#IS = 2|VU| - |E| 2-rk(A)

A B

Independent sets in a bipartite graph. 0-1 matrices weighted by

(1/2)rank (1 allowed at Auv if uv is an edge)

0 00 0

1 01 0

0 10 0

0 01 0

1 00 0

0 11 0

1 10 0

1 11 0

A different view of independent sets in bipartite graphs

Ge, Štefankovič ’09

#IS = 2|V U| - |E| 2-rk(A)

A B

Independent sets in a bipartite graph. 0-1 matrices weighted by

(1/2)rank (1 allowed at Auv if uv is an edge)

Question: Is there a polynomial-time samplerthat produces matrices A B with P(A) 2-rank(A)

Bij=0 Aij=0

(everything over the F2)

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

Goldberg,Jerrum’10: the chain is exponentially slow for some graphs.

BAD NEWS:

Ising model: assignment of spinsto sites weighted by the numberof neighbors that agree

Random cluster model: subgraphsweighted by the number of components and the number ofedges

High temperature expansion: even subgraphs weightedby the number of edges

Our inspiration (Ising model):

Fortuin-Kasteleyn

Newell M

ontroll ‘53

Random cluster model

Z(G,q,)= q(S)|S|

SE

number of connectedcomponents of (G,S)

(Tutte polynomial)Ising modelPotts modelchromatic polynomialFlow polynomial

Random cluster model

Z(G,q,)= q(S)|S|

SE

R2 model

R2(G,q,)= qrk(S)|S|

SE2

number of connectedcomponents of (G,S)

rank (over F2) of theadjacency matrix of (G,S)

(Tutte polynomial)Ising modelPotts modelchromatic polynomialFlow polynomial

MatchingsPerfect matchings

Independent sets (for bipartite only!)

More ?

R2 model’

easy if (x-1)(y-1)=1, or(1,1),(-1,-1),(0,-1),(-1,0)

#P-hard elsewhere

Tutte polynomial

easy if q{0,1} or =0, or (1/2,-1)

#P-hard elsewhere (GRH)

Complexity of exact evaluation

Ge, Štefankovič ’09Jaeger, Vertigan, Welsh ’90

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

spanning trees

R2(G,q,)= qrk (S)|S|

SE2‘

BIS

q

“high-temperature expansion”

(1-((u),(v))U{0,1} V{0,1} {u,v}E

1,1) = 10,1) = (1,0) = (0,0) = -1

where

2|E| #BIS =

“high-temperature expansion”

(1-((u),(v))U{0,1} V{0,1} {u,v}E

1,1) = 10,1) = (1,0) = (0,0) = -1

where

2|E| #BIS =

= (-1)|S| ((u),(v))SE U{0,1} V{0,1} {u,v}S

“high-temperature expansion”

(1-((u),(v))U{0,1} V{0,1} {u,v}E

1,1) = 10,1) = (1,0) = (0,0) = -1

where

2|E| #BIS =

= (-1)|S| ((u),(v))SE U{0,1} V{0,1} {u,v}S

= {0 if some v V has an odd number of neighbors in (UV,S) labeled by 1(-2)|V| otherwise

“high-temperature expansion”

2|E| #BIS =

= (-1)|S| ((u),(v))SE U{0,1} V{0,1} {u,v}S

bipartite adjacency matrix of (UV,S)

= 2|V|SE

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

= 2|V|+|U| SE

2- rank (A))2

“high-temperature expansion” – curious

f(A,) = |v| ( )|Av| 1-1+

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

1 1

2

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

But in fact:

f(A,) = f(A,)T

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|

SE2