A different view of independent sets in bipartite graphs
bipartite graphs
Qi Ge
Daniel Štefankovič
Daniel Štefankovič
University of Rochester
University of Rochester
A different view of independent sets in bipartite graphs
bipartite graphs
counting/sampling independent sets in general graphs:
polynomial time sampler for
Δ ≤ 5 (Dyer,Greenhill
(D
G
hill ’00
’00, L
Luby,Vigoda’99,
b Vi d ’99 W
Weitz’06).
it ’06)
no polynomial time sampler (unless NP=RP) for
Δ ≥ 25 (Dyer, Frieze, Jerrum ’02).
Glauber
Gl
b d
dynamics
i d
does nott mix
i iin polynomial
l
i l titime
for 6-regular bipartite graphs (example: union of 6
random matchings) (Dyer, Frieze, Jerrum ’02).
Δ = maximum degree of G
A different view of independent sets in bipartite graphs
bipartite graphs
counting/sampling independent sets in bipartite graphs:
polynomial time sampler for
Δ ≤ 5 (Dyer,Greenhill
(D
G
hill ’00
’00, L
Luby,Vigoda’99,
b Vi d ’99 W
Weitz’06).
it ’06)
no polynomial time sampler (unless NP=RP) for
Δ ≥ 25 (Dyer, Frieze, Jerrum ’02).
(max idependent set in bipartite graph ⇔ max matching)
Glauber
Gl
b d
dynamics
i d
does nott mix
i iin polynomial
l
i l titime
for 6-regular bipartite graphs (example: union of 6
random matchings) (Dyer, Frieze, Jerrum ’02).
Δ = maximum degree of G
How hard is counting/sampling independent sets in bipartite graphs?
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)
A different view of independent sets in bipartite graphs
bipartite graphs
Ge, Štefankovič ’09
Independent sets in Independent
sets in
a bipartite graph.
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
1 1
0 0
0 1
0 1
1 0
0 1
1 1
1 0
0‐1 matrices weighted by ((1/2)
/ )rank ((1 allowed at Auv
if uv is an edge)
A different view of independent sets in bipartite graphs
bipartite graphs
0 0
0 0
0 1
0 0
Ge, Štefankovič ’09
1 0
0 0
1 1
0 0
Independent sets in Independent
sets in
a bipartite graph.
#IS = 2|V∪U| - |E| ∑ 2-rk(A)
A≤B
1 0
1 0
0 0
1 0
0 1
1 0
1 1
1 0
0‐1 matrices weighted by ((1/2)
/ )rank ((1 allowed at Auv
if uv is an edge)
A different view of independent sets in bipartite graphs
bipartite graphs
0 0
1 0
Question:
0 0
1 0
Is there a polynomial-time
0 1
0 0 sampler
0 0
1 0
th t produces
that
d
matrices
t i 1 0 A ≤ 0 1
B with
ith
0 0
1 0
-rank(A)
P(A)
( )∝2
Ge, Štefankovič ’09
1 1
0 0
Independent sets in Independent
sets in
a bipartite graph.
1 1
1 0
Bij=0
0 ⇒ Aij=0
0
0‐1 matrices weighted by ((1/2)
/ )rank ((1 allowed at Auv
#IS = 2|V∪ U| - |E| ∑ 2-rk(A)
(everythingif uv is an edge)
over the F2)
A≤B
Natural MC
flip random entry +
Metropolis filter.
A = Xt with
ith random
d
(valid)
( lid)
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.
Our inspiration (Ising model):
Fortuin‐Kasteleyn
Ising model: assignment of spins
to sites weighted by the number
to sites weighted by the number
of neighbors that agree
Random cluster model: subgraphs
weighted by the number of
weighted by the number of components and the number of
edges
High temperature expansion: High
temperature expansion:
even subgraphs weighted
by the number of edges
Random cluster model
Z(G q )= ∑ qκ(S)μ|S|
Z(G,q,μ)=
S⊆E
number of connected
components
t off (G,S)
(G S)
(Tutte polynomial)
Ising model
Potts model
chromatic polynomial
Flow polynomial
Random cluster model
R2 model
(G q μ)= ∑ qrk(S)μ|S|
Z(G q )= ∑ qκ(S)μ|S| R2(G,q,μ)=
Z(G,q,μ)=
2
S⊆E
S⊆E
number of connected
components
t off (G,S)
(G S)
(Tutte polynomial)
Ising model
Potts model
chromatic polynomial
Flow polynomial
rank (over F2) of the
adjacency
dj
matrix
t i off (G
(G,S)
S)
Matchings
Perfect matchings
IIndependent
d
d t sets
t
(for bipartite only!)
More ?
Complexity of exact evaluation
R2‘ (G,q,μ)= ∑ qrk2(S)μ|S|
S⊆E
Tutte polynomial
p y
spanning trees
R2’ model
BIS
2|E|-|V|+|isolated V|
q
Jaeger, Vertigan, Welsh ’90
Ge, Štefankovič ’09
easy if (x-1)(y-1)=1
(x-1)(y-1)=1, or
(1,1),(-1,-1),(0,-1),(-1,0)
#P-hard elsewhere
easy if q∈{0,1}
q∈{0 1}
or μ=0, or (1/2,-1)
#P-hard elsewhere (GRH)
“high-temperature expansion”
∑
∑
∏
U {0 1} V→{0,1}
U→{0,1}
V {0 1}
2|E| #BIS =
{ }
{u,v}∈E
where
χ(1,1)
χ(1
1) = 1
χ(0,1) = χ(1,0) = χ(0,0) = -1
(1-χ(σ(u),σ(v))
“high-temperature expansion”
∑
∑
∏
U {0 1} V→{0,1}
U→{0,1}
V {0 1}
2|E| #BIS =
(1-χ(σ(u),σ(v))
{ }
{u,v}∈E
where
χ(1,1)
χ(1
1) = 1
χ(0,1) = χ(1,0) = χ(0,0) = -1
=
∑
(-1) ∑
∑
∏
U→{0,1}
{ , } V→{0,1}
S⊆E
⊆
{ , }
|S|
{u,v}∈S
χ(σ(u),σ(v))
“high-temperature expansion”
∑
∑
∏
U {0 1} V→{0,1}
U→{0,1}
V {0 1}
2|E| #BIS =
(1-χ(σ(u),σ(v))
{ }
{u,v}∈E
where
χ(1,1)
χ(1
1) = 1
χ(0,1) = χ(1,0) = χ(0,0) = -1
=
∑
(-1) ∑
∑
∏
U→{0,1}
{ , } V→{0,1}
S⊆E
⊆
{ , }
|S|
χ(σ(u),σ(v))
{u,v}∈S
={
0 if some v∈ V has an odd number of neighbors
in (U∪V,S) labeled by 1
(-2)|V| otherwise
“high-temperature expansion”
2|E| #BIS =
=
∑
((-1)) ∑
∑
U→{0,1}
S⊆E
|S|
V→{0,1}
= 2|V|∑
∏
χ(σ(u),σ(v))
χ(
( ), ( ))
{u,v}∈S
number of u such that uTA = 0 (mod 2)
S E
S⊆E
bipartite adjacency matrix of (U∪V,S)
=
2|V|+|U|
∑
S E
S⊆E
2- rank2 (A))
“high-temperature expansion” – curious
f(A,λ) = ∑
|v|
λ 1(
1-λ
1+λ
f(A,1) = 2rank2(A)
T
f(A 1) = f(A,1)
f(A,1)
f(A 1)
But in fact:
T
f(A,λ)
( ,λ) = f(A,λ)
( ,λ)
|Av|
) 1
Questions:
Is there a polynomial-time sampler that produces
( )?
matrices A ≤ B with P(A) ∝ 2-rank(A)
R2((G,q,μ)=
,q,μ) ∑ qrk(S)μ|S|
2
S⊆E
What other quantities does the R2 polynomial
encode ?