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Graph isomorphism and asymmetric graphs Pascal Schweitzer Ghent - - PowerPoint PPT Presentation

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Graph isomorphism and asymmetric graphs Pascal Schweitzer Ghent Graph Theory Workshop 2017 August 18th, Ghent Graph isomorphism and asymmetric graphs Pascal Schweitzer 1 / 38 asymmetry oracle graph isomorphism non-trivial automorphisms

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SLIDE 1

Graph isomorphism and asymmetric graphs

Pascal Schweitzer Ghent Graph Theory Workshop 2017 August 18th, Ghent

Graph isomorphism and asymmetric graphs Pascal Schweitzer 1 / 38

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SLIDE 2

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 3

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 4

The graph isomorphism problem

Two graphs are isomorphic if there is a bijection of vertices that preserves adjacency. Isomorphic graphs

Graph isomorphism and asymmetric graphs Pascal Schweitzer 3 / 38

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SLIDE 5

The graph isomorphism problem

Two graphs are isomorphic if there is a bijection of vertices that preserves adjacency. Isomorphic graphs

Graph isomorphism and asymmetric graphs Pascal Schweitzer 3 / 38

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SLIDE 6

The graph isomorphism problem

Two graphs are isomorphic if there is a bijection of vertices that preserves adjacency. Graph isomorphism (GI): Algorithmic task to decide whether two graphs are isomorphic. Isomorphic graphs

Graph isomorphism and asymmetric graphs Pascal Schweitzer 3 / 38

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SLIDE 7

Unknown complexity

Is there an efficient algorithm for graph isomorphism?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 4 / 38

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SLIDE 8

Unknown complexity

Is there an efficient algorithm for graph isomorphism? known GI ∈ NP GI NP-hard ⇒ SAT quasi-poly. (⇒ ETH false) GI NP-hard ⇒ PH collapses (GI ∈ co-AM) unknown GI ∈ P? Is GI NP-complete? GI ∈ co-NP? P NP-complete co-NP-complete co-AM

Graph isomorphism and asymmetric graphs Pascal Schweitzer 4 / 38

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SLIDE 9

Current status

2O(√

(n log n))

⇒ 2(log(n)c)

[Babai using Luks,Zemlyachenko] (1981) [Babai] (2015) Two major open subcases: group isomorphism (given by multiplication table) tournament isomorphism Both subcases have 2O(log(n)2)-time algorithms.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 5 / 38

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SLIDE 10

Problems equivalent to isomorphism

These following problems are polynomially equivalent: GI: the graph isomorphism problem col-GI: isomorphism problem of colored graphs ISO: isomorphism of general combinatorial objects Aut(G): compute generating set for automorphism group | Aut(G)|: determine the size of Aut(G). The graph isomorphism problem is actually the problem of detecting symmetries of combinatorial objects.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 6 / 38

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Automorphisms

An automorphism is an isomorphism from a graph to itself. The automorphism group captures the intrinsic symmetries of the graph.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 7 / 38

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SLIDE 12

Automorphisms

An automorphism is an isomorphism from a graph to itself. The automorphism group captures the intrinsic symmetries of the graph.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 7 / 38

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SLIDE 13

Automorphisms

An automorphism is an isomorphism from a graph to itself. The automorphism group captures the intrinsic symmetries of the graph.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 7 / 38

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SLIDE 14

Automorphisms

An automorphism is an isomorphism from a graph to itself. The automorphism group captures the intrinsic symmetries of the graph.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 7 / 38

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SLIDE 15

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 8 / 38

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Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 8 / 38

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SLIDE 17

1 1 4 3 1 5 1 1 4 3 1 5

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 18

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 19

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 20

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 21

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 22

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 23

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 24

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 25

Some reductions

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 26

Some reductions

GI col-GI Aut(G) | Aut(G)| G1 G2

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 27

Some reductions

GI col-GI Aut(G) | Aut(G)| G1 G2 W.l.o.g. G1, G2 connected. | Aut(G1 ∪ G2)| =

  • 2 · | Aut(G1)| · | Aut(G2)|

if G1 ∼ = G2 | Aut(G1)| · | Aut(G2)|

  • therwise.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 9 / 38

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SLIDE 28

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 29

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 30

Worst Case instances for IR algorithms

What is the running time of IR algorithms (such as nauty, or traces, bliss, saucy, conauto)?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 11 / 38

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Worst Case instances for IR algorithms

What is the running time of IR algorithms (such as nauty, or traces, bliss, saucy, conauto)?

  • In the worst case IR algorithms have exponential running time.

[Neuen, S.] (2017+)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 11 / 38

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SLIDE 32

Worst Case instances for IR algorithms

What is the running time of IR algorithms (such as nauty, or traces, bliss, saucy, conauto)?

  • In the worst case IR algorithms have exponential running time.

[Neuen, S.] (2017+)

W V G R(G) v1 v2 v3 w1 w2 w3 w4 w5 w6

a(w1) b(w1) a(w2) b(w2) a(w3) b(w3) a(w4) b(w4) a(w5) b(w5) a(w6) b(w6)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 11 / 38

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SLIDE 33

Benchmark graphs

200 400 600 800 1,000 1,200 1,400 10−2 10−1 100 101 102 103 104

number of vertices 104 sec Bliss Traces Nauty Saucy Conauto

Graph isomorphism and asymmetric graphs Pascal Schweitzer 12 / 38

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SLIDE 34

Benchmark graphs

200 400 600 800 1,000 1,200 1,400 10−2 10−1 100 101 102 103 104

number of vertices 104 sec Bliss Traces Nauty Saucy Conauto

These benchmarks are asymmetric graphs (rigid).

Graph isomorphism and asymmetric graphs Pascal Schweitzer 12 / 38

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SLIDE 35

Graph asymmetry

A graph G is called asymmetric (or rigid) if it does not have a non-trivial automorphism (i.e., | Aut(G)| = 1).

Graph isomorphism and asymmetric graphs Pascal Schweitzer 13 / 38

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SLIDE 36

Graph asymmetry

A graph G is called asymmetric (or rigid) if it does not have a non-trivial automorphism (i.e., | Aut(G)| = 1). Example:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 13 / 38

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SLIDE 37

Graph asymmetry

A graph G is called asymmetric (or rigid) if it does not have a non-trivial automorphism (i.e., | Aut(G)| = 1). Example: Graph asymmetry denoted GA is the algorithmic task to decide whether a given graph is asymmetric. (Many authors call this the graph automorphism problem.)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 13 / 38

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SLIDE 38

Absence of symmetry

Graph isomorphism and asymmetric graphs Pascal Schweitzer 14 / 38

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SLIDE 39

Absence of symmetry

  • Thm. exactly 18 minimal asymmetric graphs

Nešetˇ ril Conjecture [S., Schweitzer] (2017+)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 14 / 38

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SLIDE 40

Asymmetry vs isomorphism

GI col-GI Aut(G) | Aut(G)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 15 / 38

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SLIDE 41

Asymmetry vs isomorphism

GI col-GI Aut(G) | Aut(G)| GA GIAsym

Graph isomorphism and asymmetric graphs Pascal Schweitzer 15 / 38

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SLIDE 42

Asymmetry vs isomorphism

GI col-GI Aut(G) | Aut(G)| GA GIAsym

Graph isomorphism and asymmetric graphs Pascal Schweitzer 15 / 38

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SLIDE 43

Asymmetry vs isomorphism

GI col-GI Aut(G) | Aut(G)| GA GIAsym Open question: Is it harder to find all symmetries than to detect asymmetry?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 15 / 38

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SLIDE 44

Asymmetry vs isomorphism

GI col-GI Aut(G) | Aut(G)|

?

GA GIAsym Open question: Is it harder to find all symmetries than to detect asymmetry?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 15 / 38

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SLIDE 45

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 46

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 47

Tournaments

A tournament is an oriented complete graph. (exactly one directed edge between every pair of vertices)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 17 / 38

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SLIDE 48

Tournaments

A tournament is an oriented complete graph. (exactly one directed edge between every pair of vertices)

User:Nojhan/Wikimedia Commons/CC-BY-SA-3.0 Graph isomorphism and asymmetric graphs Pascal Schweitzer 17 / 38

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SLIDE 49

Symmetry problems for tournaments

GITour col-GITour Aut(T) | Aut(T)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 18 / 38

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SLIDE 50

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 51

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 52

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 53

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 54

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 55

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 56

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 57

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 58

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 59

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 60

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 61

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 62

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 63

Removing colors for tournaments

  • colored tournament isomorphismtournament isomorphism

col-GITour ≤p

m GITour

[Arvind, Das, Mukhopadhyay] (2010)

  • colored tournament asymmetry tournament asymmetry

col-GATour ≤p

m GATour

Graph isomorphism and asymmetric graphs Pascal Schweitzer 19 / 38

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SLIDE 64

Alternative to disjoint union for tournaments

For tournaments we cannot form the disjoint union. Instead we form the triangle tournament Tri(T1, T2).

Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

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SLIDE 65

Alternative to disjoint union for tournaments

For tournaments we cannot form the disjoint union. Instead we form the triangle tournament Tri(T1, T2).

T1 T2

Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

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SLIDE 66

Alternative to disjoint union for tournaments

For tournaments we cannot form the disjoint union. Instead we form the triangle tournament Tri(T1, T2).

T1 T2 T ′

1

T1 ∼ = T ′

1

Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

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SLIDE 67

Alternative to disjoint union for tournaments

For tournaments we cannot form the disjoint union. Instead we form the triangle tournament Tri(T1, T2).

T1 T2 T ′

1

T1 ∼ = T ′

1

Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

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SLIDE 68

Alternative to disjoint union for tournaments

For tournaments we cannot form the disjoint union. Instead we form the triangle tournament Tri(T1, T2).

T1 T2 T ′

1

T1 ∼ = T ′

1

| Aut(Tri(T1, T2))| =

  • 3 · | Aut(T1)|2 · | Aut(T2)|

if T1 ∼ = T2 | Aut(T1)|2 · | Aut(T2)|

  • therwise.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 20 / 38

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SLIDE 69

Asymmetry vs isomorphism for tournaments

GITour col-GITour Aut(T) | Aut(T)|

Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

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SLIDE 70

Asymmetry vs isomorphism for tournaments

GITour col-GITour Aut(T) | Aut(T)| GATour GIAsymTour

Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

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SLIDE 71

Asymmetry vs isomorphism for tournaments

GITour col-GITour Aut(T) | Aut(T)| GATour GIAsymTour

Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

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SLIDE 72

Asymmetry vs isomorphism for tournaments

GITour col-GITour Aut(T) | Aut(T)| GATour GIAsymTour Open question: Is it harder to find all symmetries than to detect asymmetry?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

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SLIDE 73

Asymmetry vs isomorphism for tournaments

GITour col-GITour Aut(T) | Aut(T)|

?

GATour GIAsymTour Open question: Is it harder to find all symmetries than to detect asymmetry?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

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SLIDE 74

Asymmetry vs isomorphism for tournaments

GITour col-GITour Aut(T) | Aut(T)| randomized GATour GIAsymTour Open question: Is it harder to find all symmetries than to detect asymmetry?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 21 / 38

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SLIDE 75

Main Result

Theorem There is a polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry. Thus: For tournaments finding all symmetries and detecting asymmetry are polynomially equivalent.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 22 / 38

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SLIDE 76

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 77

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 78

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 79

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 80

Sampling automorphisms

Technique 1: asymmetry test non-trivial automorphism sampler

Graph isomorphism and asymmetric graphs Pascal Schweitzer 25 / 38

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SLIDE 81

Sampling automorphisms

Technique 1: asymmetry test non-trivial automorphism sampler Strategy

  • fix more and more vertices until graph is asymmetric
  • make a copy of the graph
  • undo last fixing in copy
  • find alternative vertex to the vertex fixed last
  • find isomorphism from original to copy

Graph isomorphism and asymmetric graphs Pascal Schweitzer 25 / 38

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SLIDE 82

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 83

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 84

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 85

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 86

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 87

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 88

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 89

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 90

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 91

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 92

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 93

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 94

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 95

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 96

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 97

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 98

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 99

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 100

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 101

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 102

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 103

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 104

How to get automorphisms — Illustration

Automorphisms:

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 105

How to get automorphisms — Illustration

Automorphisms: ϕ1

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 106

How to get automorphisms — Illustration

Automorphisms: ϕ1

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 107

How to get automorphisms — Illustration

Automorphisms: ϕ1, ϕ2

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 108

How to get automorphisms — Illustration

Automorphisms: ϕ1, ϕ2, ϕ3

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 109

How to get automorphisms — Illustration

Automorphisms: ϕ1, ϕ2, ϕ3, . . .

Graph isomorphism and asymmetric graphs Pascal Schweitzer 26 / 38

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SLIDE 110

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 111

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 112

Sampling sets

A set of automorphisms M′ ⊆ Aut(G) is invariant if M′ϕ = M′ for all ϕ ∈ Aut(G).

Graph isomorphism and asymmetric graphs Pascal Schweitzer 28 / 38

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SLIDE 113

Sampling sets

A set of automorphisms M′ ⊆ Aut(G) is invariant if M′ϕ = M′ for all ϕ ∈ Aut(G). Only invariant sets of automorphisms are useful.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 28 / 38

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SLIDE 114

Sampling sets

A set of automorphisms M′ ⊆ Aut(G) is invariant if M′ϕ = M′ for all ϕ ∈ Aut(G). Only invariant sets of automorphisms are useful. Technique 2: sampling invariant subsets

Graph isomorphism and asymmetric graphs Pascal Schweitzer 28 / 38

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SLIDE 115

Sampling sets

A set of automorphisms M′ ⊆ Aut(G) is invariant if M′ϕ = M′ for all ϕ ∈ Aut(G). Only invariant sets of automorphisms are useful. Technique 2: sampling invariant subsets There is a technique to extract invariant subsets with high

  • probability. (Sample often and apply Chernoff bounds.)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 28 / 38

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SLIDE 116

Sampling sets

A set of automorphisms M′ ⊆ Aut(G) is invariant if M′ϕ = M′ for all ϕ ∈ Aut(G). Only invariant sets of automorphisms are useful. Technique 2: sampling invariant subsets There is a technique to extract invariant subsets with high

  • probability. (Sample often and apply Chernoff bounds.)

But: The number of samples required is polynomial in | Aut(G)|, which may be exponential in |G|.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 28 / 38

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SLIDE 117

Sampling sets

A set of automorphisms M′ ⊆ Aut(G) is invariant if M′ϕ = M′ for all ϕ ∈ Aut(G). Only invariant sets of automorphisms are useful. Technique 2: sampling invariant subsets There is a technique to extract invariant subsets with high

  • probability. (Sample often and apply Chernoff bounds.)

But: The number of samples required is polynomial in | Aut(G)|, which may be exponential in |G|. However, we can sample pairs of vertices lying in a common

  • rbit. There are less than n2 such pairs.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 28 / 38

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SLIDE 118

invariant suborbits

We call a partition π = {C1, . . . , Ct} of the vertices a partition into invariant suborbits if every Ci is contained in an orbit π is invariant under Aut(G) (i.e., πϕ = π for all ϕ ∈ Aut(G))

Graph isomorphism and asymmetric graphs Pascal Schweitzer 29 / 38

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SLIDE 119

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 120

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 121

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 122

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 123

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 124

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 125

Invariant suborbits — Illustration

Examples of invariant suborbits

Graph isomorphism and asymmetric graphs Pascal Schweitzer 30 / 38

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SLIDE 126

Finding invariant suborbits

Lemma Given an invariant sampler we can compute in polynomial time invariant suborbits (with high probability). Proof technique:

  • repeatedly sample ϕ ∈ Aut(G) and randomly pick pair

(x,ϕ(x)) with x ∈ V(G)

  • extract characteristic set of pairs
  • compute the transitive closure of the relation induced by pairs

Graph isomorphism and asymmetric graphs Pascal Schweitzer 31 / 38

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SLIDE 127

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

slide-128
SLIDE 128

graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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SLIDE 129

Computing with solvable groups

Theorem (Luks (1982)) For a solvable permutation group Γ on V and a graph G on vertex set V we can compute Γ ∩ Aut(G) in polynomial time. Facts: Tournaments have solvable automorphism group. Wreath products of solvable groups are solvable.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 33 / 38

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SLIDE 130

The quotient tournament

T: a tournament; π = {C1, . . . , Ct} : a partition of the vertices

Graph isomorphism and asymmetric graphs Pascal Schweitzer 34 / 38

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SLIDE 131

The quotient tournament

T: a tournament; π = {C1, . . . , Ct} : a partition of the vertices The quotient tournament T/π has the vertex set {C1, . . . , Ct}. The direction of the edge between Ci and Ck is the majority direction between Ci and Ck in T.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 34 / 38

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SLIDE 132

The quotient tournament

T: a tournament; π = {C1, . . . , Ct} : a partition of the vertices The quotient tournament T/π has the vertex set {C1, . . . , Ct}. The direction of the edge between Ci and Ck is the majority direction between Ci and Ck in T.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 34 / 38

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SLIDE 133

The quotient tournament

T: a tournament; π = {C1, . . . , Ct} : a partition of the vertices The quotient tournament T/π has the vertex set {C1, . . . , Ct}. The direction of the edge between Ci and Ck is the majority direction between Ci and Ck in T.

  • Graph isomorphism and asymmetric graphs

Pascal Schweitzer 34 / 38

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SLIDE 134

The quotient tournament

T: a tournament; π = {C1, . . . , Ct} : a partition of the vertices The quotient tournament T/π has the vertex set {C1, . . . , Ct}. The direction of the edge between Ci and Ck is the majority direction between Ci and Ck in T.

  • Note: if all Ci have odd size this operation is well defined.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 34 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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SLIDE 139

Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament) compute ∆ := Aut(T/π)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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SLIDE 140

Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament) compute ∆ := Aut(T/π) compute Θ := Aut(T[C1])

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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SLIDE 141

Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament) compute ∆ := Aut(T/π) compute Θ := Aut(T[C1]) compute Γ := Θ ≀ ∆ (wreath product)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament) compute ∆ := Aut(T/π) compute Θ := Aut(T[C1]) compute Γ := Θ ≀ ∆ (wreath product) compute Γ ∩ Aut(T)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament) compute ∆ := Aut(T/π) compute Θ := Aut(T[C1]) compute Γ := Θ ≀ ∆ (wreath product) compute Γ ∩ Aut(T) Output: Aut(T) = Γ ∩ Aut(T)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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Basic Strategy

Technique 3: invariant suborbits automorphism group Input: A tournament T; invariant suborbit oracle compute a partition π = {C1, . . . , Ct} into invariant suborbits For simplicity assume all induced subtournaments T[Ci] are isomorphic. compute T/π (quotient tournament) compute ∆ := Aut(T/π) compute Θ := Aut(T[C1]) compute Γ := Θ ≀ ∆ (wreath product) compute Γ ∩ Aut(T) Output: Aut(T) = Γ ∩ Aut(T) A more careful case distinction and some running time analysis show that the overall process runs in polynomial time.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 35 / 38

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graph isomorphism asymmetry tournaments asymmetry

  • racle

non-trivial automorphisms invariant suborbits automorphism group and isomorphism

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Summary

Theorem There is a polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry.

Graph isomorphism and asymmetric graphs Pascal Schweitzer 37 / 38

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Summary

Theorem There is a polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry. Open: How about for graphs? How about for groups?

Graph isomorphism and asymmetric graphs Pascal Schweitzer 37 / 38

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Summary

Theorem There is a polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry. Open: How about for graphs? How about for groups? Related results: Canonization: With [Arvind, Das, Mukhopadhyay] (2010) we get an

analogous result for canonization. Hardness: tournament asymmetry is hard for NL, C=L, PL, DET, and MODkL under AC0 reductions. [Wager] (2007)

Graph isomorphism and asymmetric graphs Pascal Schweitzer 37 / 38

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Cumulative Prize Money

Graph isomorphism and asymmetric graphs Pascal Schweitzer 38 / 38

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Cumulative Prize Money

Prize for a proof that GI ∈ P or GI / ∈ P!

100 Euro

Graph isomorphism and asymmetric graphs Pascal Schweitzer 38 / 38

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Cumulative Prize Money

Prize for a proof that GI ∈ P or GI / ∈ P!

+ 100 Euro

Graph isomorphism and asymmetric graphs Pascal Schweitzer 38 / 38

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Cumulative Prize Money

Prize for a proof that GI ∈ P or GI / ∈ P!

105 Euro

Graph isomorphism and asymmetric graphs Pascal Schweitzer 38 / 38