Complexity Classes and the Graph Isomorphism Problem

Complexity Classes and the Graph Isomorphism problem

Jacobo Toran

University of Ulm

NMI Complexity and Cryptography Workshop,

IIT Gandhinagar, November 2016

1

Isomorphism and Automorphism

Two graphs G1 = (V1, E1), G2 = (V2, E2)

Isomorphism: Bijection ϕ : V1 → V2,(u, v) ∈ E1 ⇔ (ϕ(u), ϕ(v)) ∈ E2.

Automorphism: Permutation ϕ : V1 → V1,(u, v) ∈ E1 ⇔ (ϕ(u), ϕ(v)) ∈ E1.

ϕ

2

Upper Bounds

G,H graphs

GI: are G and H isomorphic?

GA: does G have a non-trivial automorphism?

P

NP

GIGA

3

Upper Bounds

G,H graphs



Time(2logO(1) n)

[Babai 15]

P

NP

GIGA

4

G,H graphs



Time(2logO(1) n)

P

NP

GIGA

coAM

5

G,H graphs



Time(2logO(1) n)

P

NP

GIGA

coAM

⊕P

6

G,H graphs



Time(2logO(1) n)

P

NP

GIGA

coAM

⊕P

SPP

7

G,H graphs



Time(2logO(1) n)

P

NP

GIGA

coAM

⊕P

SPP

UAP

8

Linear Programing:

NP ∩ coNP [Gale, Kuhn Tucker 48]

P [Khachiyan 79]

Primes:

NP ∩ coNP [Pratt 75]

coRP [Solovay, Strassen 77]

nO(log logn) [Adleman, Pomerance, Rumely 81]

RP [Adleman, Huang 87]

UP ∩ coUP [Fellows, Koblitz 92]

P [Agrawal, Kayal, Saxena 02]

9

Let A and B be two groups, with B ≤ A and ϕ ∈ A.

Bϕ = {πϕ : π ∈ B}

is a subset of A called a right coset of B in A.

Two right cosets Bϕ1, Bϕ2 are either disjoint or equal.

A = Bϕ1 +Bϕ2 + · · ·+Bϕk.

The set {ϕ1, . . . , ϕk} is called a complete right transversal for B in A.

10

Let G1 and G2 be two graphs.

Fact: If G1 and G2 are isomorphic then, Iso(G1, G2) is a right coset of

Aut(G1), and thus |Iso(G1, G2)| = |Aut(G1)|.

11

Permutation groups

Let A ≤ Sn. For i ∈ {1, . . . , n} the orbit of i in A is the set

{j : ∃ϕ ∈ A, ϕ(i) = j}.

A[i 7→j], is the set of all permutations in A which map i to j.

For X ⊆ {1, . . . , n} i ∈ {1, . . . , n} the pointwise stabilizer of X in A.

A[X] is the set of permutations

A[X] = {ϕ ∈ A : ∀x ∈ X, ϕ(x) = x}.

A[i] = A[{i}] = {ϕ ∈ A : ϕ(i) = i}.

12

Graph G = (V,E), X ⊆ V ,

Aut(G)[X] = Aut(G[X]).

GX

G[X]

G[1,...,i] := G(i)

13

Pointwise stabilizers play an important role in group theoretic algorithms.

One can construct a “tower” of stabilizers in the following way: let

Xi = {1, . . . , i} and denote A[Xi] by A(i), then

{id} = A(n) ≤ A(n−1) ≤ · · · ≤ A(1) ≤ A(0) = A.

14

Lemma:

Let i ∈ {1, . . . n} and A = A[i]π1 +A[i]π2 + · · ·+A[i]πd.

Then d is the size of the orbit of i in A, and for all ψ ∈ A[i]π,

π(i) = ψ(i).

If {j1, . . . , jd} is the orbit of i in A, then

A is partitioned into d sets of equal cardinality:

|A[i 7→j1]| = · · · = |A[i 7→jd]|.

Thus, |A| = d ∗ |A[i]|, and for every j ∈ {1, . . . , n}, the number of

permutations in A which map i to j is either 0 or |A[i]|.

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Corollary:

Let di be the size of the orbit of i in A(i−1), 1 ≤ i ≤ n.

|A| =n∏

i=1

di

.

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j is in the orbit of i in G(i−1) iff the graphs are isomorphic

j

[i− 1]

i i

[i− 1]

j

Corollary: [Mathon 79] #Aut(G) and #Iso(G1, G2) can be computed

with non-adaptive queries to GI.

17

Theorem: [Sims 70], [Furst,Hopcroft,Luks 80]

Let Ti be a complete right transversal for A(i) in A(i−1), 1 ≤ i ≤ n, and

let K0 =⋃n

i=1 Ti, then

1. every element π ∈ A can be expressed uniquely as a product

π = ϕ1ϕ2 . . . ϕn with ϕi ∈ Ti,

2. from K0 membership in A can be tested in O(n2) steps,

3. the order of A can be determined in O(n2) steps.

K0 is called a strong generator set for A.

18

Complexity classesCounting classes

For a nondeterministic poly-time machine M and x ∈ Σ∗,

accM (x) is the number of accepting computation paths of M on input x.

(accAM (x) for an oracle set A.)

rejM (x) is the number of rejecting computation paths of M on input x.

gapM (x) = accM (x)− rejM (x)

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Class of all languages L for which there exists a nondeterministic

polynomial-time machine M such that, for all x,

NP x ∈ L =⇒ accM (x) > 0,

x 6∈ L =⇒ accM (x) = 0.

UP x ∈ L =⇒ accM (x) = 1,

x 6∈ L =⇒ accM (x) = 0.

ModkP x ∈ L =⇒ accM (x) ≡ 1mod k,

x 6∈ L =⇒ accM (x) 6≡ 1 mod k.

20

Tournament GI is in Mod2P [Goldschlager, Parberry 86]

If G is tournament graph, id is the only automorphism π ∈ Aut(G) that

is self-inverse (π = π−1).

If π 6= id, then for some i, π(i) 6= i. But π(π(i)) = i and this implies

(i, π(i)) ∈ E ⇔ (π(i), i) ∈ E.

All nontrivial automorphisms of a tournament graph G can be grouped into

pairs (πi, π−1i ) and |Aut(G)| is odd.

#GI(G,H) =

#GA(G), G ≡ H

0, G 6≡ H

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A function f : Σ∗ → N is in #P if there is a nondeterministic

polynomial-time machine M such that for every x in Σ∗,

f(x) = accM (x).

A function f : Σ∗ → Z is in Gap-P if there is a nondeterministic

polynomial-time machine M such that for every x in Σ∗,

f(x) = gapM (x).

⇐⇒ ∃f1, f2 ∈ #P, ∀x ∈ Σ∗, f(x) = f1(x)− f2(x).

22

Class of all languages L for which there exists a nondeterministic

polynomial-time machine M such that, for all x,

C=P x ∈ L =⇒ gapM (x) = 0,

x 6∈ L =⇒ gapM (x) > 0.

SPP x ∈ L =⇒ gapM (x) = 1,

x 6∈ L =⇒ gapM (x) = 0.

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GA ∈ SPP [Kobler,Schoning,T. 92]

Let G = (V,E), |V | = n. Consider the function

f(G) =∏

1≤i<j≤n

(|Aut(G)[i]| − |Aut(G)[i 7→j]|).

f ∈ Gap-P.

If G ∈ GA then there is a non-trivial automorphism π ∈ Aut(G) such

that π(i) = j for some i, j, i < j.

π ∈ Aut(G)[i 7→j] and |Aut(G)[i]| = |Aut(G)[i 7→j]|.

At least one of the factors of f(G) and f(G) = 0.

If G 6∈ GA, then ∀i, j, i < j, |Aut(G)[i 7→j]| = 0. and

Aut(G)[i] = {id}.

All the factors of f(G) have value 1 and f(G) = 1.

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Theorem: Let L ⊆ Σ⋆. If L can be computed in polynomial time by

asking only such queries y to an oracle set A ∈ NP for which

accMA(y) ∈ {0, 1} (UP like queries), then L ∈ SPP .

MA nondeterministic poly-time machine computing A.

25

Theorem: [Kobler,Schoning,T. 92]

GI can be computed in polynomial time by asking only such queries y to

an oracle set A ∈ NP for which accMA(y) ∈ {0, n!}.


A = {(G1, G2,m) | G1 ≡ G2,m ∈ N}

MA on input (G1, G2,m) guesses x, y and accepts iff x is an

isomorphism and 1 ≤ y ≤ m.

accMA(G1, G2,m) = m ·#GI(G1, G2),

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input (G,H);

d := 1; /∗ d is used to count the number of automorphisms of G ∗/

for i := n downto 1 do

di := 1; /∗ di determines the size of the orbit of i in Aut(G)(i−1) ∗/

for j := i+ 1 to n do

/∗ test whether j lies in the orbit of i in Aut(G)(i−1) ∗/

G1 := G(i−1)[i] ; G2 := G

(i−1)[j] ;

if (G1, G2, n!/d) ∈ A then di := di + 1 end;

end;

d := d ∗ di; /∗ now d = #GA(G(i−1)) ∗/

end

if (G,H, n!/d) ∈ A then accept else reject end;

27

The queries (G1, G2,m) are of the form (G(i−1)[i] , G

(i−1)[j] , n!/d).

There are either 0 or #GA(G(i)) many automorphisms in Aut(G(i−1))

such that ϕ(i) = j.

Since d = #GA(G(i)), it follows that #GI(G1, G2) ∈ {0, d} and

accMA(G1, G2, n!/d) ∈ {0, n!}.

28

GI ∈ SPP [Arvind, Kurur 02]

Theorem: There is a poly-time algorithm that on input a generator set for

B ≤ Sn and π ∈ Sn computes the lexicographically least element of

Bπ.

Theorem: GI can be computed in polynomial time by asking only such

queries y to an oracle set A ∈ NP for which accMA(y) ∈ {0, 1}.


29

A = {(S, i, j, π) | S ⊆ Aut(G(i)) and π is a partial permutation

that fixes [i− 1] and π(i) = j

and there is a ϕ ∈ Aut(G(i−1)) such that

ϕ extends π and ϕ = lex− least(〈S〉ϕ)}

A ∈ NP

If 〈S〉 = Aut(G(i−1)) then there is at most one such extension ϕ of π.

30

Games with unique winning strategies

Alternating generalization of UP

Def: [Niedermeier Rossmanith 98] An alternating Turing machine is

unambiguous if every accepting existential configuration has exactly one

move to an accepting configuration, and every rejecting universal

configuration has exactly one move to a rejecting configuration. The class

UAP contists of all languages accepted by polynomial-time unambiguous

alternating machines.

UAP ⊆ SPP

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Theorem: [Crasmaru, Glaßer, Regan, Sangupta 04]

The class of problems that can be computed in polynomial time with

“UP-like” queries to an oracle set in NP is contained in UAP.

Corollary: GI ∈ UAP.

32

Complexity classesProbabilistic classes

Def: L ∈ NP if there is a set D ∈ P and a polynomial p, for all x,

x ∈ L⇔ [∃y, |y| = p(|x|) : 〈x, y〉 ∈ D]

Def: [Babai 85] L ∈ AM if there is a set D ∈ P and polynomial p such

that for all x,

x ∈ L⇒ Probw∈Σp(|x|) [∃y, |y| = p(|x|) : 〈x, y, w〉 ∈ D] ≥ 34 ,

x 6∈ L⇒ Probw∈Σp(|x|) [∃y, |y| = p(|x|) : 〈x, y, w〉 ∈ D] ≤ 14 ,

AM = BP·NP

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Standard AM method for isomorphims problems

Given two structures A,B

1) Define a set S(A,B) in NP and a function f ∈ FP s.t.

A 6≡ B ⇒ ||S(A,B)|| ≥ 2f(n).

A ≡ B ⇒ ||S(A,B)|| ≤ f(n).

2) Estimate the size of ||S(A,B)|| using randomized hashing.

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Given graphs G1, G2 on n vertices, consider the set

N(G1, G2) = {(H,ϕ) | (H ≡ G1 or H ≡ G2) and ϕ ∈ Aut(H)}

= {(H,ϕ) | H ≡ G1 and ϕ ∈ Aut(H)} ∪

{(H,ϕ) | H ≡ G2 and ϕ ∈ Aut(H)}

|N(G1, G2)| =

n! if G1 ≡ G2

2 · n! if G1 6≡ G2

35

Derandomization of AM protocols

Hardness hypothesis imply AM = NP

Hardness hypothesis:

[Arvind Kobler 97] ∀ǫ > 0 there is a language L in NE ∩ coNE so that

any nondeterministic circuit of of size 2ǫn agrees with Ln on at most

1/2 + 2−ǫn inputs.

Open: Is there a better way for derandomizing the AM protocol for Graph

non-Iso than derandomizing the whole class AM?

36

The Minimum Circuit Size Problem (MCSP)

MCSP: [Yablonski 59] Given s ∈ N and a Boolean function f on n

variables, represented by its truth table of size 2n , determine if f has a

circuit of size s.

MCSP ∈ NP.

Not known to be NP-complete.

If MCSP is NP-complete (under logspace reductions) then P 6= PSPACE

[Murray, Williams 15]

GI is randomly reducible to MCSP [Allender, Das 14]

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PGI

bb

b b

b

b

NP coNP

coAM

UP

b

b

b

UAP

SPP

ModkP

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Lower Bounds

Is GI hard for some complexity class?

What problems can be reduced to GI?

Hard instances of GI

(bench marks, concrete algorithms, proof systems ...)

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Undirected Graph (non)Reachability is reducible to GI

s tG

Is t reachable from s?

s1 t1

G′

s2 t2

There is no path from s to t in G ⇐⇒

there is some automorphism mapping s1 to s2 in G′

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there is some automorphism mapping s1 to s2 in G′ ⇐⇒

G′′ is isomorphic to H ′′.

s1 t1

G′′

s2 t2

s3 t3

H ′′

s4 t4

⇒ GI is hard for LOGSPACE

41

Basic multiplication gadget

Let A = {1, . . .m} be finite group, x, y, z ∈ A.

Simulation of x · y = z

⊙

y

x

z

42

Basic multiplication gadget

y

x

z⊙

b

b

b

b

b

bb

b

b

b

b

b

x1xi. . .

xm

y1

yj

. . .

ymum,m

ui,j

u1,1

. . .

. . .z1·1 = z1

zi·j

zm

. . .

. . .

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Simulation of an addition gate in Z2

y

x

z⊕

b

b

b

b

b

b

b

b

b

b

x0

x1

y0

y1

u1,1

u1,0

u0,1

u0,0

z0

z1

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Automorphisms of the gadgetCase 1: A Abelian group

Element a ∈ A is encoded as the mapping i→ a · i in the corresponding

vertices.

Level 1:

i→ x · i on the subgraph from input x.

j → y · j on the subgraph from input y.

Level 2:

(i, j) → (x · i, y · j) on the middle vertices.

Level 3:

k = i · j → x · i · y · j = x · y · i · j = x · y · k

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Simulation of a circuit with gates over A

⊕

⊕

⊕

⊕

⊕

1

0

1

1

1 ·

···

··G2

·

···

··G2

G2

··

··G2

G2·· ··

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CFI-graphs [Cai, Furer, Immerman 92] are constructed following this

principle.

The evaluation of circuits over Abelian groups is as hard as DET, the class

of problems reducible to the Determinant.

GI is hard for DET [T 04].

AC0 ⊆ NC1 ⊆ L ⊆ NL ⊆ DET ⊆ NC2 ⊆. . . P

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Automorphisms of the gadgetCase 2: A non-Abelian

y

x

z⊙

b

b

b

b

b

bb

b

b

b

b

b

x1xi. . .

xm

y1

yj

. . .

ymum,m

ui,j

u1,1

. . .

. . .z1·1 = z1

zi·j

zm

. . .

. . .

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Automorphisms of the gadgetCase 2: A non-Abelian

Representation of the group elements:

We associate each element k ∈ A with the set of pairs (i, j) satisfying

k = i · j.

Element k ∈ A is encoded as the mapping i→ l · i · r for a pair (l, r)

with l · r = k in the corresponding vertices. Many encodings for a are

possible.

L R

Res

⊙

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Lemma: For every pair of elements l, r ∈ A, the automorphisms that

map each vertex k in a Res subgraph to l · k · r are those that for some

element t map each vertex i in subgraph L to l · i · t, and maps each

vertex j in subgraph R to t−1 · j · r.

(The elements encoded in subgraphs L and R are respectively l · t and

t−1 · r.)

(l, t) (t−1, r)

(l, r)

⊙

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Tree-Evaluation

Let A be a finite group. Given a binary tree T with a group element a ∈ A

in each leaf, is the product of the elements equal to id?

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For every finite (quasi) group, the Tree-Evaluation Problem is reducible to

GI.

(l, r)

(t−1, r)(l, t)

a b

a · b = l · r

a = l · t

⇒ t−1 · r = a−1 · l · r = b

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Tree Evaluation is in NC1

Tree Evaluation over S5 is NC1-complete

Circuit Evaluation over S5 is P-complete

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b

b

b

NC1

L

NL

b

bb

SAC1

AC1

DET

NC2 b

PGI

bb

54

b

b

b

NC1

L

NL

b

bb

SAC1

AC1

DET

NC2 b

PGI

bb

b b

b

b

NP coNP

coAM

UP

b

b

b

UAP

SPP

ModkP

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Complexity Classes and the Graph Isomorphism Problem

Engineering

Transcript of Complexity Classes and the Graph Isomorphism Problem