A graph polynomial for independent sets of
bipartite graphs
Qi Ge
Department of Computer Science
University of Rochester
joint work with Daniel Štefankovič
EaGL Ⅲ
Graph Polynomials
Algebraic method for analyzing properties of
graph
Some graph polynomials
Chromatic polynomial
Reliability polynomial
Tutte polynomial
Tutte polynomial
A polynomial in two variables
Random cluster model
Z(G; q, µ) =
�
q
κ(S) |S|
µ
S⊆E
sum over weighted subgraphs (V, S)
each edge has weight µ
each connected component has weight q
Tutte polynomial (cont.)
C3
3
2
3
2
Z(C3 ; q, µ) = q + 3µq + (µ + 3µ )q
3
qµ
q µ
2
q
2
q µ
qµ
2
2
qµ
2
3
q µ
qµ
2
A new graph polynomial R2
The R2 polynomial of a graph G = (V, E)
R2 (G; q, µ) =
�
q rk2 (S) µ|S|
S⊆E
sum over weighted subgraphs (V, S)
each edge has weight µ
rk2 (S) is the rank of the adjacency matrix of
(V, S) over F2
A new graph polynomial R2 (cont.)
C3
3
2
2
R2 (C3 ; q, µ) = (µ + 3µ + 3µ)q + 1

0
0
0

0
0
0

0
1
1
0
0
1

0
1
0
1
0
0

1
0
0
2
q µ
2 2
q µ
0
0
0

0
0
0

0
1
1
1
2
q µ
2 2
q µ
1
0
1

1
1
0
2 3
q µ
2
q µ
2 2
q µ
A new graph polynomial R2 (cont.)
R2 (G; q, µ) =
�
q rk2 (S) µ|S|
S⊆E
Properties of the R2 polynomial
number of matchings P (G; 0)
P (G; µ) = R2 (G; µ
−1/2
, µ) =
�
µ|S|−rk2 (S)/2
S⊆E
number of perfect matchings
P1 (G; t, µ) = t|V | R2 (G; 1/t, µ)
P2 (G; 0)
P2 (G; µ) = µ−|V |/2 P1 (G; 0, µ)
number of independent sets if the graph is
bipartite
Polynomial
�
R2
�
R2
The
polynomial of a bipartite graph
G = (U ∪ W, E)
�
R2 (G; λ, µ)
=
�
λ
rk2 (S) |S|
µ
S⊆E
each edge has weight µ
rk2 (S) is the rank of the bipartite adjacency
matrix of (U ∪ W, S) over F2
Polynomial
K2,2
�
R2
(cont.)
R2� (K2,2 ; λ, µ) = 1 + λµ4 + 2λ2 µ2 + 4λµ2 + 4λ2 µ3 + 4λµ
1
�
�
�
1 1
0 0
�
λµ
2
λµ4
λ2 µ 2
λ2 µ 2
2
2
2
λµ
λµ
λµ
0 1
1 1
�
λ µ
λ µ
λ µ
λ µ
1 0
0 0
�
λµ
λµ
λµ
λµ
2 3
2 3
2 3
2 3
Polynomial
(cont.)
�
R2
For bipartite graphs
R2 (G; λ, µ) = R2� (G; λ2 , µ)
(Main Theorem) Let G = (U ∪ W, E) be a
bipartite graph. The number of independent sets
of G is given by
|U |+|W |−|E|
2
�
R2 (G; 1/2, 1)
Main Theorem
Q = {(u, B) | u ∈ F22 , B ∈ M2,2 (F2 ), B ≤ A, uT B ≡ 0
A=
�
�
1 0
1 1
� �
1
u=
1
� �
1
u=
0
B=
�
B=
�
(mod 2)}
1 0
1 0
�
∈Q
1 0
1 0
�
�∈ Q
Main Theorem
Q = {(u, B) | u ∈ F22 , B ∈ M2,2 (F2 ), B ≤ A, uT B ≡ 0
A=
�
�
1 0
1 1
(mod 2)}
P(B)?
B=
�
P(B) =
1
1
0
0
� �
1
u=
1
�
� �
0
u=
0
�
u:uT B≡0
(mod 2)
1
2n1 −rk2 (B)
2−rk2 (B)
=
= �
|Q|
|Q|
R2 (G; 1/2, 1)
Main Theorem
Q = {(u, B) | u ∈ F22 , B ∈ M2,2 (F2 ), B ≤ A, uT B ≡ 0
A=
�
�
1 0
1 1
P(u)?
half choices
� �
1
u=
0
full choices
P(u) =
(mod 2)}
�
B:uT B≡0
(mod 2)
B=
B=
�
�
0 0
0 0
0 0
1 0
�
�
B=
B=
�
�
0
0
0
1
1
2#1 (A)−(n2 −k)
2k
=
=
|Q|
|Q|
#BIS(G)
0
1
0
1
�
�
Main Theorem
Q = {(u, B) | u ∈ F22 , B ∈ M2,2 (F2 ), B ≤ A, uT B ≡ 0
A=
�
�
1 0
1 1
P(B) =
�
(mod 2)
1
2n1 −rk2 (B)
2−rk2 (B)
=
= �
|Q|
|Q|
R2 (G; 1/2, 1)
�
(mod 2)
1
2#1 (A)−(n2 −k)
2k
=
=
|Q|
|Q|
#BIS(G)
u:uT B≡0
P(u) =
B:uT B≡0
|Q| = 2n1 R2� (G; 1/2, 1) = 2#1 (A)−n2 #BIS(G)
(mod 2)}
Complexity of exact evaluation of
R2� (G; λ, µ) =
FP
λ ∈ {0, 1}
µ=0
�
�
R2
λrk2 (S) µ|S|
S⊆E
µ
(λ, µ) = (1/2, −1)
#P-hard
λ = 1/2 and µ �∈ {0, −1}
#P-hard assuming GRH
λ �∈ {0, 1, 1/2} and µ �= 0
0
−1
1
λ
Approximate sampling problem
Rank Weighted Subgraphs RWS(λ, µ)
instance: a bipartite graph G = (U ∪ W, E)
output: S ⊆ E with probability
P(S) ∝ λ
rk2 (S) |S|
µ
RWS(λ, µ)
Markov chain Monte Carlo method
proper Markov chain
mixing time
Sing bond flip chain for RWS(λ, µ)
1. pick an edge e ∈ E at random
2. let S = Xt ⊕ {e}
3. set Xt+1 = S with probability
(1/2) min{1, λrk2 (S)−rk2 (Xt ) µ|S|−|Xt | }
and Xt+1 = Xt with the remaining probability
RWS(λ, µ) (cont.)
Question: for which classes of bipartite
graphs does the single bond flip chain mix
(in polynomial time)?
The single bond flip chain for trees mixes in
polynomial time
rk2 is the size of the maximum matching
mixing time can be bounded by using the
canonical paths method
RWS(λ, µ) (cont.)
Question: for which classes of bipartite
graphs does the single bond flip chain mix
(in polynomial time)?
There exists bipartite graphs for which the
single bond flip chain needs exponential time
to mix for λ = 1/2 and µ = 1 [Goldberg &
Jerrum 2010]
How about grids, planar graphs?
Thanks !