a graph polynomial for independent sets of bipartite graphsa new graph polynomial the polynomial of...
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joint work with Daniel Štefankovič
A graph polynomial for independent sets of bipartite graphs
Qi GeDepartment of Computer Science
University of Rochester
EaGL Ⅲ
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Graph Polynomials
Algebraic method for analyzing properties of graph
Some graph polynomials
Chromatic polynomial
Reliability polynomial
Tutte polynomial
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Tutte polynomial
A polynomial in two variables
Random cluster model
sum over weighted subgraphs
each edge has weight
each connected component has weight
Z(G; q, µ) =�
S⊆E
qκ(S)µ|S|
q
µ
(V, S)
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Tutte polynomial (cont.)
q3
q2µq2µ q2µ
qµ2qµ2 qµ2
qµ3
Z(C3; q, µ) = q3 + 3µq2 + (µ3 + 3µ2)qC3
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A new graph polynomial
The polynomial of a graph
sum over weighted subgraphs
each edge has weight
is the rank of the adjacency matrix of over
R2
R2 G = (V,E)
R2(G; q, µ) =�
S⊆E
qrk2(S)µ|S|
µ
rk2(S)
(V, S) F2
(V, S)
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A new graph polynomial (cont.)R2
C3
q2µ q2µq2µ
0 0 00 0 10 1 0
0 1 11 0 11 1 0
q2µ3
0 0 00 0 00 0 0
1
0 1 11 0 01 0 0
q2µ2 q2µ2 q2µ2
R2(C3; q, µ) = (µ3 + 3µ2 + 3µ)q2 + 1
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A new graph polynomial (cont.)
Properties of the polynomialnumber of matchings
number of perfect matchings
number of independent sets if the graph is bipartite
R2
R2
P (G;µ) = R2(G;µ−1/2, µ) =�
S⊆E
µ|S|−rk2(S)/2
P (G; 0)
P2(G; 0)
P1(G; t, µ) = t|V |R2(G; 1/t, µ) P2(G;µ) = µ−|V |/2P1(G; 0, µ)
R2(G; q, µ) =�
S⊆E
qrk2(S)µ|S|
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Polynomial R�2
The polynomial of a bipartite graph
each edge has weight
is the rank of the bipartite adjacency matrix of over
R�2
G = (U ∪W,E)
R�2(G;λ, µ) =
�
S⊆E
λrk2(S)µ|S|
µ
rk2(S)
(U ∪W,S) F2
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Polynomial (cont.)R�2
K2,2
λµλµλµ
λ2µ3λ2µ3λ2µ3
λµ2λµ2λµ2
�1 00 0
�
λ2µ3�0 11 1
�
λµ2�1 10 0
�
λ2µ2 λ2µ2λµ41
R�2(K2,2;λ, µ) = 1 + λµ4 + 2λ2µ2 + 4λµ2 + 4λ2µ3 + 4λµ
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Polynomial (cont.)R�2
For bipartite graphs
(Main Theorem) Let be a bipartite graph. The number of independent sets of is given by
G = (U ∪W,E)
G
2|U |+|W |−|E|R�2(G; 1/2, 1)
R2(G;λ, µ) = R�2(G;λ2, µ)
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Main Theorem
A =
�1 01 1
�
Q = {(u,B) |u ∈ F22, B ∈ M2,2(F2), B ≤ A, uTB ≡ 0 (mod 2)}
u =
�11
�B =
�1 01 0
�∈ Q
B =
�1 01 0
�u =
�10
��∈ Q
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Main Theorem
A =
�1 01 1
�
Q = {(u,B) |u ∈ F22, B ∈ M2,2(F2), B ≤ A, uTB ≡ 0 (mod 2)}
u =
�11
�
B =
�1 01 0
�
u =
�00
�
P(B)?
P(B) =�
u:uTB≡0 (mod 2)
1
|Q| =2n1−rk2(B)
|Q| =2−rk2(B)
R�2(G; 1/2, 1)
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Main Theorem
A =
�1 01 1
�
Q = {(u,B) |u ∈ F22, B ∈ M2,2(F2), B ≤ A, uTB ≡ 0 (mod 2)}
P(u)?
P(u) =�
B:uTB≡0 (mod 2)
1
|Q| =2#1(A)−(n2−k)
|Q| =2k
#BIS(G)
B =
�0 00 0
�
u =
�10
�
full choices
half choicesB =
�0 00 1
�
B =
�0 01 0
�B =
�0 01 1
�
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Main Theorem
A =
�1 01 1
�
Q = {(u,B) |u ∈ F22, B ∈ M2,2(F2), B ≤ A, uTB ≡ 0 (mod 2)}
P(u) =�
B:uTB≡0 (mod 2)
1
|Q| =2#1(A)−(n2−k)
|Q| =2k
#BIS(G)
P(B) =�
u:uTB≡0 (mod 2)
1
|Q| =2n1−rk2(B)
|Q| =2−rk2(B)
R�2(G; 1/2, 1)
|Q| = 2n1R�2(G; 1/2, 1) = 2#1(A)−n2#BIS(G)
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Complexity of exact evaluation of
FP
#P-hard
#P-hard assuming GRH
R�2
λ ∈ {0, 1}µ = 0
(λ, µ) = (1/2,−1)
λ = 1/2 and µ �∈ {0,−1}
λ �∈ {0, 1, 1/2} and µ �= 0
R�2(G;λ, µ) =
�
S⊆E
λrk2(S)µ|S|
λ
µ
0 1
−1
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Approximate sampling problem
Rank Weighted Subgraphs
instance: a bipartite graph
output: with probability
RWS(λ, µ)
G = (U ∪W,E)
S ⊆ E
P(S) ∝ λrk2(S)µ|S|
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Sing bond flip chain for 1. pick an edge at random2. let3. set with probability
and with the remaining probability
e ∈ E
S = Xt ⊕ {e}Xt+1 = S
RWS(λ, µ)
(1/2)min{1,λrk2(S)−rk2(Xt)µ|S|−|Xt|}Xt+1 = Xt
RWS(λ, µ)
Markov chain Monte Carlo methodproper Markov chainmixing time
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Question: for which classes of bipartite graphs does the single bond flip chain mix (in polynomial time)?
(cont.)
The single bond flip chain for trees mixes in polynomial time
is the size of the maximum matching
mixing time can be bounded by using the canonical paths method
RWS(λ, µ)
rk2
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Question: for which classes of bipartite graphs does the single bond flip chain mix (in polynomial time)?
(cont.)RWS(λ, µ)
There exists bipartite graphs for which the single bond flip chain needs exponential time to mix for and [Goldberg & Jerrum 2010]
How about grids, planar graphs?
λ = 1/2 µ = 1
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Thanks !