combinatorics - tcs @ njutcs.nju.edu.cn/slides/comb2017/polya.pdf · graph isomorphism (gi)...
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Combinatorics南京大学尹一通
Pólya’s Theory of Counting
George Pólya(1887-1985)
Counting with SymmetryRotation :
Rotation & Reflection:
Symmetries
Symmetry
01
23
4
5
rotation
reflection
configuration x : [n] ! [m] X = [m][n]
positions colors
permutation ⇡ : [n]1-1���!
on-to
[n]
Permutation Groups(G, ·)group · : G⇥G ! Gwith binary operator
• closure:• associativity:• identity:• inverse:
⇡,� 2 G ) ⇡ · � 2 G⇡ · (� · ⌧) = (⇡ · �) · ⌧
� = ⇡�18⇡ 2 G, 9� 2 G,⇡ · � = � · ⇡ = e9e 2 G, 8⇡ 2 G, e · ⇡ = ⇡
commutative (abelian) group: ⇡ · � = � · ⇡
symmetric groupcyclic groupDihedral group
Sn
Cn
Dn
: all permutations: rotations
: rotations & reflections
Permutation Groupssymmetric group
cyclic group
Dihedral group
Sn
Cn
Dn
: all permutations
: rotations
: rotations & reflections
⇡ : [n]1-1���!
on-to
[n]
h(012 · · ·n� 1)i generated by (012 · · ·n� 1)
⇡(i) = (i+ 1) mod n⇡ = (012 · · ·n� 1)
generated by (012 · · ·n� 1) and
⇢(i) = (n� 1)� i
⇢
Group Action0
1
23
4
5
01
23
4
5
configuration x : [n] ! [m]
permutation ⇡ : [n]1-1���!
on-to
[n]
group action
X = [m][n]
group G
� : G⇥X ! X
• associativity:• identity:
(⇡ · �) � x = ⇡ � (� � x)e � x = x
(⇡ � x)(i) = x(⇡(i))
Graph Isomorphism (GI) Problem
?
• GI is in NP, but is NOT known to be in P or NPC
• trivial algorithm: O(n!) time
• Babai-Luks ’83: time
nvertices
Babai 2015: a quasi-polynomial time algorithm!npolylog(n) = 2polylog(n)
2O(pn logn)
⇠=
input: two undirected graphs G and Houtput: ?G ⇠= H
input: two strings a permutation group output:
String Isomorphism (SI)
⇠= ?
x, y : [n] ! [m]
G ✓ Sn
x
⇠=G y? (9� 2 G s.t. � � x = y)
String Isomorphism (SI)
a graph X(V,E) is a string
x :
✓V
2
◆! {0, 1}
all permutations on V induces a permutation group on
on vertex pairs
two graphs X ⇠= Y iff their string versions x
⇠=S(2)Vy
positions ⟷ vertex-pairs
1: edge0: no edge
Johnson group:
input: two strings a permutation group output:
x, y : [n] ! [m]
G ✓ Sn
x
⇠=G y? (9� 2 G s.t. � � x = y)
S(2)V ⇢ S(V2)
✓V
2
◆
01
23
4
5
01
23
4
5
group action � : G⇥X ! X
Orbits
orbit of x :Gx = {⇡ � x | ⇡ 2 G}
equivalent class of configuration x
X/G = {Gx | x 2 X}
our goal: count |X/G|
X = [2][n]
G = Cn
G = Dn
X/G :
X/G :
2 ×4 ×
configuration x : [n] ! [m]
~v = (n1, n2, . . . , nm)
a~v : # of config. (up to symmetry) with ni many color i
n1 + n2 + · · ·+ nm = n
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
s.t.
a(2,4) = 3
a~v : # of config. (up to symmetry) with ni many color i
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
G = Dn X/G :
= y61 + y51y2 + 3y41y22 + 3y31y
32 + 3y21y
42 + y1y
52 + y62FD6(y1, y2)
a~v : # of config. (up to symmetry) with ni many color i
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
FG(y1, y2, . . . , ym) = PG
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!Pólya’s enumeration formula (1937,1987):
⇡ =
`1z }| {(· · · )
`2z }| {(· · · ) · · ·
`kz }| {(· · · )| {z }
k cycles
M⇡(x1, x2, . . . , xn) =kY
i=1
x`i
PG(x1, x2, . . . , xn) =1
|G|X
⇡2G
M⇡(x1, x2, . . . , xn)cycle index:
FD20(r, q, l)
X = [2][n]
G = Cn
G = Dn
X/G :
X/G :
Burnside’s Lemmagroup action � : G⇥X ! X
orbit of x :Gx = {⇡ � x | ⇡ 2 G}
invariant set of π :
X⇡ = {x 2 X | ⇡ � x = x}
|X/G| = 1
|G|X
⇡2G
|X⇡|
Burnside’s Lemma:
X/G = {Gx | x 2 X}
group action � : G⇥X ! X
orbit of x :Gx = {⇡ � x | ⇡ 2 G}
invariant set of π :X⇡ = {x 2 X | ⇡ � x = x}
X/G = {Gx | x 2 X}
stabilizer of x :G
x
= {⇡ 2 G | ⇡ � x = x}
8x 2 X, |Gx
||Gx| = |G|Lemma:
A(⇡, x) =
(1 ⇡ � x = x
0 otherwise
|X⇡
| =X
x2X
A(⇡, x) |Gx
| =X
⇡2G
A(⇡, x)
A(⇡, x) =
(1 ⇡ � x = x
0 otherwise
|X⇡
| =X
x2X
A(⇡, x) |Gx
| =X
⇡2G
A(⇡, x)
double counting:
8x 2 X, |Gx
||Gx| = |G|Lemma:
= |G|X
x2X
1
|Gx|
X
⇡2G
|X⇡| =X
⇡2G
X
x2X
A(⇡, x) =X
x2X
X
⇡2G
A(⇡, x) =X
x2X
|Gx
|
= |G|X
x2X
1
|Gx|
X
⇡2G
|X⇡|
= |G||X/G|X
i=1
X
x2Xi
1
|Gx|
orbits:
= |G||X/G|X
i=1
X
x2Xi
1
|Xi
|
= |G||X/G|X
i=1
1 = |G||X/G|
X1, X2, . . . , X|X/G|
a partition of X
|X/G| = 1
|G|X
⇡2G
|X⇡| Burnside’sLemma
Burnside’s Lemmagroup action � : G⇥X ! X
orbit of x :Gx = {⇡ � x | ⇡ 2 G}
invariant set of π :
X⇡ = {x 2 X | ⇡ � x = x}
|X/G| = 1
|G|X
⇡2G
|X⇡|
Burnside’s Lemma:
X/G = {Gx | x 2 X}
Cycle Decompositionpermutation ⇡ : [n]
1-1���!on-to
[n]
✓0 1 2 · · · n� 1
⇡(0) ⇡(1) ⇡(2) · · · ⇡(n� 1)
◆
✓0 1 2 3 42 4 0 1 3
◆= (0 2)(1 4 3)
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Cycle Decompositionpermutation ⇡ : [n]
1-1���!on-to
[n]
⇡ � x = x
invariant every cycle of π has the same color
8i 2 [n], x(⇡(i)) = x(i)
�� � *�1%"*�1& 0�*�$�7&+"� 8+IP�!?=��
;� �� ;<� ;� <� ;X� �
>� =� >� =�>�7�XXXX�� �1�� &1��
;XX� X<� <� ;�
;� �B�F� &�< ���X� >@F� 0�:� =� X>� =� =F� E�� � ��1�� �L�� ��
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⇡ � x = x
invariant every cycle of π has the same color
8i 2 [n], x(⇡(i)) = x(i)
X = [m][n]
⇡ = (· · · )(· · · ) · · · (· · · )| {z }k cycles
m-colorings of n positions
|X⇡| = |{x 2 X | ⇡ � x = x}| = mk
|X/G| = 1
|G|X
⇡2G
|X⇡|
Burnside’s Lemma:
=1
|G|X
⇡2G
m#cycle(⇡)
X = [2][n]
G = Cn
G = Dn
X/G :
X/G :
2 ×4 ×
configuration x : [n] ! [m]
~v = (n1, n2, . . . , nm)
a~v : # of config. (up to symmetry) with ni many color i
n1 + n2 + · · ·+ nm = n
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
s.t.
a(2,4) = 3
a~v : # of config. (up to symmetry) with ni many color i
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
G = Dn X/G :
= y61 + y51y2 + 3y41y22 + 3y31y
32 + 3y21y
42 + y1y
52 + y62FD6(y1, y2)
Cycle index
⇡ =
`1z }| {(· · · )
`2z }| {(· · · ) · · ·
`kz }| {(· · · )| {z }
k cycles
permutation ⇡ : [n]1-1���!
on-to
[n]
monomial:
from group G
PG(x1, x2, . . . , xn) =1
|G|X
⇡2G
M⇡(x1, x2, . . . , xn)
cycle index:
M⇡(x1, x2, . . . , xn) =kY
i=1
x`i
Burnside’s Lemma:PG(m,m, . . . ,m| {z }
n
) =1
|G|X
⇡2G
M⇡(m,m, . . . ,m| {z }n
)|X/G| =
a~v : # of config. (up to symmetry) with ni many color i
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
FG(y1, y2, . . . , ym) = PG
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!Pólya’s enumeration formula (1937,1987):
⇡ =
`1z }| {(· · · )
`2z }| {(· · · ) · · ·
`kz }| {(· · · )| {z }
k cycles
M⇡(x1, x2, . . . , xn) =kY
i=1
x`i
PG(x1, x2, . . . , xn) =1
|G|X
⇡2G
M⇡(x1, x2, . . . , xn)cycle index:
G. Pólya (1937). Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta
Mathematica 68 (1): 145–254.
G. Pólya; R. C. Read (1987). Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer-Verlag.
~v = (n1, n2, . . . , nm) n1 + n2 + · · ·+ nm = n
X = [m][n] X
~v = {x 2 [m][n] | 8i 2 [m], x�1(i) = ni}
Burnside’s Lemma:
a~v =1
|G|X
⇡2G
|X~v⇡ |
X
~v⇡ = {x 2 X
~v | ⇡ � x = x}invariant set of π :
a(2,4) = 3
X(2,4) :
=X
~v=(n1,...,nm)n1+···+nm=n
|X~v⇡ |y
n11 yn2
2 · · · ynmm
⇡ =
`1z }| {(· · · )
`2z }| {(· · · ) · · ·
`kz }| {(· · · )| {z }
k cycles
X
~v⇡ = {x 2 X
~v | ⇡ � x = x}
⇡ � x = x
invariant every cycle of π has the same color8i 2 [n], x(⇡(i)) = x(i)
M⇡(x1, x2, . . . , xn) =kY
i=1
x`irecall:
M⇡
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!
=
(y`11 + y`12 + · · ·+ y`1m)(y`21 + y`22 + · · ·+ y`2m) · · · (y`k1 + y`k2 + · · ·+ y`km )
=X
~v=(n1,...,nm)n1+···+nm=n
|X~v⇡ |y
n11 yn2
2 · · · ynmm
M⇡
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!
X
~v⇡ = {x 2 X
~v | ⇡ � x = x}
Burnside’s Lemma:
a~v =1
|G|X
⇡2G
|X~v⇡ |
pattern inventory :
=X
~v=(n1,...,nm)n1+···+nm=n
1
|G|X
⇡2G
|X~v⇡ |!yn11 yn2
2 · · · ynmm
=1
|G|X
⇡2G
X
~v=(n1,...,nm)n1+···+nm=n
|X~v⇡ |y
n11 yn2
2 · · · ynmm
=1
|G|X
⇡2G
M⇡
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!
FG(y1, y2, . . . , ym)
=X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
= PG
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!cycle index :
a~v : # of config. (up to symmetry) with ni many color i
pattern inventory :
FG(y1, y2, . . . , ym) =X
~v=(n1,...,nm)n1+···+nm=n
a~v yn11 yn2
2 · · · ynmm
(multi-variate) generating function
⇡ =
`1z }| {(· · · )
`2z }| {(· · · ) · · ·
`kz }| {(· · · )| {z }
k cycles
M⇡(x1, x2, . . . , xn) =kY
i=1
x`i
PG(x1, x2, . . . , xn) =1
|G|X
⇡2G
M⇡(x1, x2, . . . , xn)cycle index:
= PG
mX
i=1
yi,mX
i=1
y2i , . . . ,mX
i=1
yni
!
FG(y1, y2, . . . , ym)
Pólya’s enumeration formula (1937,1987):
FD20(r, q, l)
Orbits of Group Actions