Structural and Logical Approaches to the Graph Isomorphism Problem - - PowerPoint PPT Presentation

Structural and Logical Approaches to the Graph Isomorphism Problem

Martin Grohe

RWTH Aachen

Graph Isomorphism (GI) Given two graphs, decide if they are isomorphic.

2

Graph Isomorphism (GI) Given two graphs, decide if they are isomorphic.

2

Graph Isomorphism (GI) Given two graphs, decide if they are isomorphic.

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

2

Status of the Problem

GI is in NP, but not known to be in PTIME or NP-complete.

3

Status of the Problem

GI is in NP, but not known to be in PTIME or NP-complete.

◮ One of the few natural problems with this property

3

Status of the Problem

GI is in NP, but not known to be in PTIME or NP-complete.

◮ One of the few natural problems with this property ◮ GI was studied in the chemistry literature in the 1950s

3

Status of the Problem

GI is in NP, but not known to be in PTIME or NP-complete.

◮ One of the few natural problems with this property ◮ GI was studied in the chemistry literature in the 1950s ◮ Open problem in [Karp, 1972] and [Garey and Johnson, 1979]

3

Status of the Problem

GI is in NP, but not known to be in PTIME or NP-complete.

◮ One of the few natural problems with this property ◮ GI was studied in the chemistry literature in the 1950s ◮ Open problem in [Karp, 1972] and [Garey and Johnson, 1979] ◮ Can be solved fairly well in practice.

3

This Talk

• 1. A Brief Survey
• 2. Colour Refinement and Weisfeiler-Lehman
• 3. A Linear Programming Approach to Graph Isomorphism
• 4. Concluding Remarks

4

A Brief Survey

5

Complexity

6

Complexity

GI is unlikely to be NP-complete:

Theorem (Boppana, Hastad, Zachos 1987; Schöning 1988)

If GI is NP-complete then the polynomial hierarchy collapses to its second level.

6

Complexity

GI is unlikely to be NP-complete:

Theorem (Boppana, Hastad, Zachos 1987; Schöning 1988)

If GI is NP-complete then the polynomial hierarchy collapses to its second level.

Theorem (Toran 2004)

GI is hard for the class DET (and hence for NL).

6

Upper Bounds

Worst Case (Zemlyachenko; Babai 1981; Babai, Luks 1983)

GI can be solved in time 2O(√

n·log n).

7

Upper Bounds

Worst Case (Zemlyachenko; Babai 1981; Babai, Luks 1983)

GI can be solved in time 2O(√

n·log n).

Average Case (Babai, Erdös, Selkow 1980)

GI can be solved in expected linear time (in the G(n, 1/2) model).

7

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

◮ bounded eigenvalue multiplicities

Babai et al. 1982

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

◮ bounded eigenvalue multiplicities

Babai et al. 1982

◮ bounded degree

Luks 1982

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

◮ bounded eigenvalue multiplicities

Babai et al. 1982

◮ bounded degree

Luks 1982

◮ graphs with excluded minors

Ponomarenko 1988

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

◮ bounded eigenvalue multiplicities

Babai et al. 1982

◮ bounded degree

Luks 1982

◮ graphs with excluded minors

Ponomarenko 1988

◮ bounded tree width

Bodlaender 1990

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

◮ bounded eigenvalue multiplicities

Babai et al. 1982

◮ bounded degree

Luks 1982

◮ graphs with excluded minors

Ponomarenko 1988

◮ bounded tree width

Bodlaender 1990

◮ interval graphs

in AC2 Klein 1996 in logspace Köbler et al. 2010

8

Tractable Classes

GI can be solved in polynomial time when restricted to classes of:

◮ planar graphs

Hopcroft, Tarjan 1972 in linear time Hopcroft, Wong 1974 in logspace Datta et al. 2009

◮ bounded genus

Filotti, Mayer 1980; Miller 1980

◮ bounded eigenvalue multiplicities

Babai et al. 1982

◮ bounded degree

Luks 1982

◮ graphs with excluded minors

Ponomarenko 1988

◮ bounded tree width

Bodlaender 1990

◮ interval graphs

in AC2 Klein 1996 in logspace Köbler et al. 2010

◮ graphs with excluded topological subgraphs

G., Marx 2012

8

paths trees planar bounded genus bounded tree width excluded minor excluded top. subgraph bounded degree interval graphs

9

Hard Classes

GI restricted to the following classes is as hard as the general problem:

◮ bipartite graphs ◮ chordal graphs ◮ rectangle intersection graphs (Uehara 2008) ◮ graphs of bounded degeneracy ◮ graphs of bounded expansion

10

paths trees planar bounded genus bounded tree width excluded minors excluded top. subgraphs bounded degree interval graphs bounded expansion bounded degeneracy rectangle intersection graphs

11

Algorithms

GI-algorithms can be divided into three groups:

◮ graph theoretic algorithms ◮ group theoretic algorithms ◮ combinatorial algorithms

12

Graph Theoretic Algorithms

The algorithms for the following classes are typical graph algorithms exploiting the graph theoretic properties of the classes:

13

Graph Theoretic Algorithms

The algorithms for the following classes are typical graph algorithms exploiting the graph theoretic properties of the classes:

◮ planar graphs (Hopcroft, Tarjan 1972; Hopcroft, Wong 1974;

Datta et al. 2009)

◮ bounded genus (Filotti, Mayer 1980; Miller 1980) ◮ bounded tree width (Bodlaender 1990) ◮ interval graphs (Klein 1996; Köbler et al. 2010)

13

Group Theoretic Algorithms

The following algorithms are based on computing generators for the automorphism groups; they exploit properties of the automorphism groups:

14

Group Theoretic Algorithms

The following algorithms are based on computing generators for the automorphism groups; they exploit properties of the automorphism groups:

◮ coloured graphs with bounded colour-class-size (Babai 1979

randomized; Furst, Hopcroft Luks 1980 deterministic)

◮ graphs with bounded eigenvalue multiplicities (Babai,

Grigoriev, Mount 1982)

◮ graphs of bounded degree (Luks 1982) ◮ k-contractible graphs (Miller 1983) ◮ graphs with excluded minors (Ponomarenko 1988) ◮ general graphs in time 2O(√ n·log n) (Zemlyachenko; Babai

1981; Babai, Luks 1983)

14

Group Theoretic Algorithms

The following algorithms are based on computing generators for the automorphism groups; they exploit properties of the automorphism groups:

◮ coloured graphs with bounded colour-class-size (Babai 1979

randomized; Furst, Hopcroft Luks 1980 deterministic)

◮ graphs with bounded eigenvalue multiplicities (Babai,

Grigoriev, Mount 1982)

◮ graphs of bounded degree (Luks 1982) ◮ k-contractible graphs (Miller 1983) ◮ graphs with excluded minors (Ponomarenko 1988) ◮ general graphs in time 2O(√ n·log n) (Zemlyachenko; Babai

1981; Babai, Luks 1983) The group theoretic approach dominated research on the graph isomorphism problem since the early 1980s.

14

Combinatorial Algorithms

Colour Refinement Weisfeiler Lehman Individualisation Refinement

15

Combinatorial Algorithms

Colour Refinement Weisfeiler Lehman Individualisation Refinement

Simple and generic algorithms that do not use the properties of specific graph classes.

15

Combinatorial Algorithms

Colour Refinement Weisfeiler Lehman Individualisation Refinement

Simple and generic algorithms that do not use the properties of specific graph classes. Most practical GI-tools, for example Nauty (McKay 1981), are based on Individualisation Refinement.

15

Colour Refinement and Weisfeiler-Lehman

16

Colour Refinement

Iteratively compute colouring of vertices of graph G

17

Colour Refinement

Iteratively compute colouring of vertices of graph G Initialisation All vertices get the same colour.

17

Colour Refinement

Iteratively compute colouring of vertices of graph G Initialisation All vertices get the same colour. Refinement Step Two nodes v, w get different colours if there is some colour c such that v and w have different numbers of neighbours of colour c.

17

Colour Refinement

Iteratively compute colouring of vertices of graph G Initialisation All vertices get the same colour. Refinement Step Two nodes v, w get different colours if there is some colour c such that v and w have different numbers of neighbours of colour c. Refinement is repeated until colouring stays stable.

Start 17

Colour Refinment as an Isomorphism Test

To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G, H.

18

Colour Refinment as an Isomorphism Test

To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G, H.

◮ Colour refinement distinguishes G and H if there is a colour c

• f the stable colouring such that G and H have different

numbers of vertices of colour c.

18

Colour Refinment as an Isomorphism Test

To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G, H.

◮ Colour refinement distinguishes G and H if there is a colour c

• f the stable colouring such that G and H have different

numbers of vertices of colour c.

◮ Colour refinement identifies G if it distinguishes G from all

• ther graphs.

18

Colour Refinment as an Isomorphism Test

To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G, H.

◮ Colour refinement distinguishes G and H if there is a colour c

• f the stable colouring such that G and H have different

numbers of vertices of colour c.

◮ Colour refinement identifies G if it distinguishes G from all

• ther graphs.

Examples

• 1. Colour refinement identifies all forests (Immerman, Lander

1990).

18

Colour Refinment as an Isomorphism Test

To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G, H.

◮ Colour refinement distinguishes G and H if there is a colour c

• f the stable colouring such that G and H have different

numbers of vertices of colour c.

◮ Colour refinement identifies G if it distinguishes G from all

• ther graphs.

Examples

• 1. Colour refinement identifies all forests (Immerman, Lander

1990).

• 2. Colour refinement identifies almost all graphs (Babai, Erdös,

Selkow 1980).

18

Colour Refinment as an Isomorphism Test

To use colour refinement as an isomorphism test, apply it to disjoint union of the input graphs G, H.

◮ Colour refinement distinguishes G and H if there is a colour c

• f the stable colouring such that G and H have different

numbers of vertices of colour c.

◮ Colour refinement identifies G if it distinguishes G from all

• ther graphs.

Examples

• 1. Colour refinement identifies all forests (Immerman, Lander

1990).

• 2. Colour refinement identifies almost all graphs (Babai, Erdös,

Selkow 1980).

• 3. Colour refinement does not distinguish any two regular graphs

with the same number of vertices and the same number of edges.

18

Running Time

n := |V (G)|, m := |E(G)|

◮ Stable colouring is reached after at most n refinement steps.

19

Running Time

n := |V (G)|, m := |E(G)|

◮ Stable colouring is reached after at most n refinement steps. ◮ Stable colouring can be computed in time O((n + m) log n)

(Cardon, Crochemore 1982, also see Paige, Tarjan 1985)

19

Running Time

n := |V (G)|, m := |E(G)|

◮ Stable colouring is reached after at most n refinement steps. ◮ Stable colouring can be computed in time O((n + m) log n)

(Cardon, Crochemore 1982, also see Paige, Tarjan 1985)

◮ Ω((n + m) log n) lower bound for natural class of algorithms

(algorithms that iteratively refine partitions) (Bonsma, Berkholz, G. 2013)

19

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

20

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

◮

v ∼ = w if the mapping vi → wi is an isomorphism between the induced subgraphs

20

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

◮

v ∼ = w if the mapping vi → wi is an isomorphism between the induced subgraphs

◮

v and w are i-neighbours if vj = wj for all j = i

20

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

◮

v ∼ = w if the mapping vi → wi is an isomorphism between the induced subgraphs

◮

v and w are i-neighbours if vj = wj for all j = i

The k-dimensional Weisfeiler-Lehman algorithm (k-WL)

Given G, it iteratively computes colouring of V (G)k

20

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

◮

v ∼ = w if the mapping vi → wi is an isomorphism between the induced subgraphs

◮

v and w are i-neighbours if vj = wj for all j = i

The k-dimensional Weisfeiler-Lehman algorithm (k-WL)

Given G, it iteratively computes colouring of V (G)k Initialisation v, w get different colours if v ∼ = w.

20

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

◮

v ∼ = w if the mapping vi → wi is an isomorphism between the induced subgraphs

◮

v and w are i-neighbours if vj = wj for all j = i

The k-dimensional Weisfeiler-Lehman algorithm (k-WL)

Given G, it iteratively computes colouring of V (G)k Initialisation v, w get different colours if v ∼ = w. Refinement Step v, w get different colours if there is a colour c and an i ≤ k such that v and w have different numbers of i-neighbours of colour c.

20

The Weisfeiler-Lehman Algorithm

Notation

• v = (v1, . . . , vk),

w = (w1, . . . , wk) vertex tuples.

◮

v ∼ = w if the mapping vi → wi is an isomorphism between the induced subgraphs

◮

v and w are i-neighbours if vj = wj for all j = i

The k-dimensional Weisfeiler-Lehman algorithm (k-WL)

Given G, it iteratively computes colouring of V (G)k Initialisation v, w get different colours if v ∼ = w. Refinement Step v, w get different colours if there is a colour c and an i ≤ k such that v and w have different numbers of i-neighbours of colour c. Refinement is repeated until colouring stays stable.

20

Remarks

Running time

k-WL on n-vertex graphs G.

◮ Stable colouring is reached after at most nk steps. ◮ Stable colouring can be computed in time O(nk log n).

21

Remarks

Running time

k-WL on n-vertex graphs G.

◮ Stable colouring is reached after at most nk steps. ◮ Stable colouring can be computed in time O(nk log n).

Weisfeiler-Lehman vs Colour Refinement

For all G, H: 2-WL distinguishes G, H ⇐ ⇒ Colour refinement distinguishes G, H.

21

The Strength of Weisfeiler-Lehman

◮ 3-WL identifies almost all regular graphs (Bollobás 1982)

22

The Strength of Weisfeiler-Lehman

◮ 3-WL identifies almost all regular graphs (Bollobás 1982) ◮ There is a k such that k-WL identifies all interval graphs

(Laubner 2010)

22

The Strength of Weisfeiler-Lehman

◮ 3-WL identifies almost all regular graphs (Bollobás 1982) ◮ There is a k such that k-WL identifies all interval graphs

(Laubner 2010)

◮ For every class C of graphs with excluded minors there is a k

such that k-WL identifies all graphs in C (G. 2011)

22

The Strength of Weisfeiler-Lehman

◮ 3-WL identifies almost all regular graphs (Bollobás 1982) ◮ There is a k such that k-WL identifies all interval graphs

(Laubner 2010)

◮ For every class C of graphs with excluded minors there is a k

such that k-WL identifies all graphs in C (G. 2011)

◮ For every k there are nonisomorphic 3-regular graphs Gk, Hk of

size O(k) that are not distinguished by k-WL (Cai, Fürer, Immerman 1982)

22

paths trees planar bounded genus bounded tree width excluded minors excluded top. minors bounded degree interval graphs bounded eigenvalue multiplicities bounded colour classes

Weisfeiler-Lehman group theoretic structural

23

A Linear Programming Approach to Graph Isomorphism

24

Integer Linear Program for GI

G, H n-vertex graphs with vertex sets V , W and adjacency matrices A = (avv′) ∈ {0, 1}V ×V , B = (bww′) ∈ {0, 1}W ×W .

25

Integer Linear Program for GI

G, H n-vertex graphs with vertex sets V , W and adjacency matrices A = (avv′) ∈ {0, 1}V ×V , B = (bww′) ∈ {0, 1}W ×W .

Observation

G and H are isomorphic iff there is a permutation matrix X such that X TAX = B,

25

Integer Linear Program for GI

G, H n-vertex graphs with vertex sets V , W and adjacency matrices A = (avv′) ∈ {0, 1}V ×V , B = (bww′) ∈ {0, 1}W ×W .

Observation

G and H are isomorphic iff there is a permutation matrix X such that X TAX = B, or equivalently AX = XB. (⋆)

25

Integer Linear Program for GI

G, H n-vertex graphs with vertex sets V , W and adjacency matrices A = (avv′) ∈ {0, 1}V ×V , B = (bww′) ∈ {0, 1}W ×W .

Observation

G and H are isomorphic iff there is a permutation matrix X such that X TAX = B, or equivalently AX = XB. (⋆) ILP-Formulation

• v′∈V

avv′xv′w =

• w′∈W

xvw′bw′w

n

• w∈W

xvw =

n

• v∈V

xvw = 1 xvw ∈ {0, 1} for all v ∈ V , w ∈ W .

25

LP-Relaxation

• v′∈V

avv′xv′w =

• w′∈W

xvw′bw′w

• w∈W

xvw =

• v∈V

xvw = 1 xvw ≥ 0 for all v ∈ V , w ∈ W .

26

LP-Relaxation

• v′∈V

avv′xv′w =

• w′∈W

xvw′bw′w

• w∈W

xvw =

• v∈V

xvw = 1 xvw ≥ 0 for all v ∈ V , w ∈ W .

Theorem (Tinhofer 1991)

Colour refinement distinguishes G and H iff the linear program has no solution.

26

Sherali-Adams Hierarchy

Level-k Sherali-Adams relaxation (Lk) Variables xp for p ⊆ V × W with |p| ≤ k.

27

Sherali-Adams Hierarchy

Level-k Sherali-Adams relaxation (Lk) Variables xp for p ⊆ V × W with |p| ≤ k.

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w

27

Sherali-Adams Hierarchy

Level-k Sherali-Adams relaxation (Lk) Variables xp for p ⊆ V × W with |p| ≤ k.

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w x∅ = 1

• w∈W

xp∪vw =

• v∈V

xp∪vw = xp for all |p| ≤ k − 1, v, w

27

Sherali-Adams Hierarchy

Level-k Sherali-Adams relaxation (Lk) Variables xp for p ⊆ V × W with |p| ≤ k.

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w x∅ = 1

• w∈W

xp∪vw =

• v∈V

xp∪vw = xp for all |p| ≤ k − 1, v, w xp ≥ 0 for all p

27

Sherali-Adams Hierarchy

Level-k Sherali-Adams relaxation (Lk) Variables xp for p ⊆ V × W with |p| ≤ k.

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w x∅ = 1

• w∈W

xp∪vw =

• v∈V

xp∪vw = xp for all |p| ≤ k − 1, v, w xp ≥ 0 for all p

Theorem (Atserias and Maneva 2012)

• 1. If k-WL distinguishes G and H, then Lk has no solution.
• 2. If Lk has a no solution, then (k + 1)-WL distinguishes G and

H.

27

A Closer Look

Theorem (Atserias and Maneva 2012)

• 1. If k-WL distinguishes G and H, then Lk has no solution.
• 2. If Lk has a no solution, then (k + 1)-WL distinguishes G and

H.

28

A Closer Look

Theorem (Atserias and Maneva 2012)

• 1. If k-WL distinguishes G and H, then Lk has no solution.
• 2. If Lk has a no solution, then (k + 1)-WL distinguishes G and

H.

Theorem (G., Otto 2012)

• 1. There are graphs G, H such that k-WL does not distinguish G

and H and Lk has no solution.

28

A Closer Look

Theorem (Atserias and Maneva 2012)

• 1. If k-WL distinguishes G and H, then Lk has no solution.
• 2. If Lk has a no solution, then (k + 1)-WL distinguishes G and

H.

Theorem (G., Otto 2012)

• 1. There are graphs G, H such that k-WL does not distinguish G

and H and Lk has no solution.

• 2. There are graphs G, H such that Lk has a solution and

(k + 1)-WL distinguishes G and H.

28

An Exact Correspondence

COMPk

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w for all |p| ≤ k − 1, v, w

29

An Exact Correspondence

COMPk

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w for all |p| ≤ k − 1, v, w CONTk x∅ = 1

• w∈W

xp∪vw =

• v∈V

xp∪vw = xp for all |p| ≤ k − 1, v, w

29

An Exact Correspondence

COMPk

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w for all |p| ≤ k − 1, v, w CONTk x∅ = 1

• w∈W

xp∪vw =

• v∈V

xp∪vw = xp for all |p| ≤ k − 1, v, w NNk Xp ≥ 0 for all |p| ≤ k

29

An Exact Correspondence

COMPk

• v′∈V

avv′xp∪v′w =

• w′∈W

xp∪vw′bw′w for all |p| ≤ k − 1, v, w CONTk x∅ = 1

• w∈W

xp∪vw =

• v∈V

xp∪vw = xp for all |p| ≤ k − 1, v, w NNk Xp ≥ 0 for all |p| ≤ k

Theorem (G., Otto 2012)

k-WL distinguishes G and H iff COMPk−1 ∪ CONTk ∪ NNk has a solution.

29

Concluding Remarks

30

Is GI in solvable in polynomial time?

31

Is GI in solvable in polynomial time?

Why not?

31

Is GI in solvable in polynomial time?

Why not?

◮ We have good reasons to believe that GI is not NP-complete.

Almost all natural problems in NP that are not NP-complete are in PTIME.

31

Is GI in solvable in polynomial time?

Why not?

◮ We have good reasons to believe that GI is not NP-complete.

Almost all natural problems in NP that are not NP-complete are in PTIME.

◮ It is not a strong argument that despite considerable efforts

noone has found a polnomial time algorithms. It took a long time for PRIMALITY as well. Furthermore, for some problems (like k-DISJOINT PATHS) we only know extremely complicated algorithms relying on a deep theory.

31

Is GI in solvable in polynomial time?

Why not?

◮ We have good reasons to believe that GI is not NP-complete.

Almost all natural problems in NP that are not NP-complete are in PTIME.

◮ It is not a strong argument that despite considerable efforts

noone has found a polnomial time algorithms. It took a long time for PRIMALITY as well. Furthermore, for some problems (like k-DISJOINT PATHS) we only know extremely complicated algorithms relying on a deep theory.

◮ There are complexity theoretic results (“hardness vs

derandomisation for AM”) indicating that at least GI is in co-NP.

31