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.
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Status of the Problem
GI is in NP, but not known to be in PTIME or NP-complete.
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Status of the Problem
GI is in NP, but not known to be in PTIME or NP-complete.
I
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.
I
One of the few natural problems with this property
I
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.
I
One of the few natural problems with this property
I
GI was studied in the chemistry literature in the 1950s
I
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.
I
One of the few natural problems with this property
I
GI was studied in the chemistry literature in the 1950s
I
Open problem in [Karp, 1972] and [Garey and Johnson, 1979]
I
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:
I
planar graphs
in linear time
in logspace
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
I
bounded genus
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
Filotti, Mayer 1980; Miller 1980
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
I
bounded genus
I
bounded eigenvalue multiplicities
Filotti, Mayer 1980; Miller 1980
Babai et al. 1982
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
I
bounded genus
I
bounded eigenvalue multiplicities
I
bounded degree
Filotti, Mayer 1980; Miller 1980
Babai et al. 1982
Luks 1982
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
I
bounded genus
I
bounded eigenvalue multiplicities
I
bounded degree
I
graphs with excluded minors
Filotti, Mayer 1980; Miller 1980
Babai et al. 1982
Luks 1982
Ponomarenko 1988
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
I
bounded genus
I
bounded eigenvalue multiplicities
I
bounded degree
I
graphs with excluded minors
I
bounded tree width
Filotti, Mayer 1980; Miller 1980
Babai et al. 1982
Luks 1982
Ponomarenko 1988
Bodlaender 1990
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
I
bounded genus
I
bounded eigenvalue multiplicities
I
bounded degree
I
graphs with excluded minors
I
bounded tree width
I
interval graphs
in AC2
in logspace
Filotti, Mayer 1980; Miller 1980
Babai et al. 1982
Luks 1982
Ponomarenko 1988
Bodlaender 1990
Klein 1996
Köbler et al. 2010
8
Tractable Classes
GI can be solved in polynomial time when restricted to classes of:
I
planar graphs
in linear time
in logspace
I
bounded genus
I
bounded eigenvalue multiplicities
I
bounded degree
I
graphs with excluded minors
I
bounded tree width
I
interval graphs
in AC2
in logspace
I
Hopcroft, Tarjan 1972
Hopcroft, Wong 1974
Datta et al. 2009
Filotti, Mayer 1980; Miller 1980
graphs with excluded topological subgraphs
Babai et al. 1982
Luks 1982
Ponomarenko 1988
Bodlaender 1990
Klein 1996
Köbler et al. 2010
G., Marx 2012
8
excluded top. subgraph
excluded minor
bounded
degree
bounded
genus
interval graphs
bounded
tree width
planar
trees
paths
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Hard Classes
GI restricted to the following classes is as hard as the general
problem:
I
bipartite graphs
I
chordal graphs
I
rectangle intersection graphs (Uehara 2008)
I
graphs of bounded degeneracy
I
graphs of bounded expansion
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bounded degeneracy
rectangle
intersection graphs
bounded expansion
excluded top. subgraphs
excluded minors
bounded
degree
bounded
genus
interval graphs
bounded
tree width
planar
trees
paths
11
Algorithms
GI-algorithms can be divided into three groups:
I
graph theoretic algorithms
I
group theoretic algorithms
I
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:
I
planar graphs (Hopcroft, Tarjan 1972; Hopcroft, Wong 1974;
Datta et al. 2009)
I
bounded genus (Filotti, Mayer 1980; Miller 1980)
I
bounded tree width (Bodlaender 1990)
I
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:
I
coloured graphs with bounded colour-class-size (Babai 1979
randomized; Furst, Hopcroft Luks 1980 deterministic)
I
graphs with bounded eigenvalue multiplicities (Babai,
Grigoriev, Mount 1982)
I
graphs of bounded degree (Luks 1982)
I
k-contractible graphs (Miller 1983)
I
graphs with excluded minors (Ponomarenko 1988)
√
general graphs in time 2O( n·log n) (Zemlyachenko; Babai
1981; Babai, Luks 1983)
I
14
Group Theoretic Algorithms
The following algorithms are based on computing generators for the
automorphism groups; they exploit properties of the automorphism
groups:
I
coloured graphs with bounded colour-class-size (Babai 1979
randomized; Furst, Hopcroft Luks 1980 deterministic)
I
graphs with bounded eigenvalue multiplicities (Babai,
Grigoriev, Mount 1982)
I
graphs of bounded degree (Luks 1982)
I
k-contractible graphs (Miller 1983)
I
graphs with excluded minors (Ponomarenko 1988)
√
general graphs in time 2O( n·log n) (Zemlyachenko; Babai
1981; Babai, Luks 1983)
I
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.
I Colour refinement distinguishes G and H if there is a colour c
of 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.
I Colour refinement distinguishes G and H if there is a colour c
of the stable colouring such that G and H have different
numbers of vertices of colour c.
I Colour refinement identifies G if it distinguishes G from all
other 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.
I Colour refinement distinguishes G and H if there is a colour c
of the stable colouring such that G and H have different
numbers of vertices of colour c.
I Colour refinement identifies G if it distinguishes G from all
other 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.
I Colour refinement distinguishes G and H if there is a colour c
of the stable colouring such that G and H have different
numbers of vertices of colour c.
I Colour refinement identifies G if it distinguishes G from all
other 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.
I Colour refinement distinguishes G and H if there is a colour c
of the stable colouring such that G and H have different
numbers of vertices of colour c.
I Colour refinement identifies G if it distinguishes G from all
other 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 )|
I
Stable colouring is reached after at most n refinement steps.
19
Running Time
n := |V (G )|, m := |E (G )|
I
Stable colouring is reached after at most n refinement steps.
I
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 )|
I
Stable colouring is reached after at most n refinement steps.
I
Stable colouring can be computed in time O((n + m) log n)
(Cardon, Crochemore 1982, also see Paige, Tarjan 1985)
I
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.
∼
I ~
~ if the mapping vi 7→ wi is an isomorphism between the
v =w
induced subgraphs
20
The Weisfeiler-Lehman Algorithm
Notation
~v = (v1 , . . . , vk ), w
~ = (w1 , . . . , wk ) vertex tuples.
∼
I ~
~ if the mapping vi 7→ wi is an isomorphism between the
v =w
induced subgraphs
I
~v and w
~ are i-neighbours if vj = wj for all j 6= i
20
The Weisfeiler-Lehman Algorithm
Notation
~v = (v1 , . . . , vk ), w
~ = (w1 , . . . , wk ) vertex tuples.
∼
I ~
~ if the mapping vi 7→ wi is an isomorphism between the
v =w
induced subgraphs
I
~v and w
~ are i-neighbours if vj = wj for all j 6= 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.
∼
I ~
~ if the mapping vi 7→ wi is an isomorphism between the
v =w
induced subgraphs
I
~v and w
~ are i-neighbours if vj = wj for all j 6= i
The k-dimensional Weisfeiler-Lehman algorithm (k-WL)
Given G , it iteratively computes colouring of V (G )k
~ get different colours if ~v 6∼
~.
Initialisation ~v , w
=w
20
The Weisfeiler-Lehman Algorithm
Notation
~v = (v1 , . . . , vk ), w
~ = (w1 , . . . , wk ) vertex tuples.
∼
I ~
~ if the mapping vi 7→ wi is an isomorphism between the
v =w
induced subgraphs
I
~v and w
~ are i-neighbours if vj = wj for all j 6= i
The k-dimensional Weisfeiler-Lehman algorithm (k-WL)
Given G , it iteratively computes colouring of V (G )k
~ get different colours if ~v 6∼
~.
Initialisation ~v , w
=w
~ get different colours if there is a colour c
Refinement Step ~v , w
~ have different
and an i ≤ k such that ~v and w
numbers of i-neighbours of colour c.
20
The Weisfeiler-Lehman Algorithm
Notation
~v = (v1 , . . . , vk ), w
~ = (w1 , . . . , wk ) vertex tuples.
∼
I ~
~ if the mapping vi 7→ wi is an isomorphism between the
v =w
induced subgraphs
I
~v and w
~ are i-neighbours if vj = wj for all j 6= i
The k-dimensional Weisfeiler-Lehman algorithm (k-WL)
Given G , it iteratively computes colouring of V (G )k
~ get different colours if ~v 6∼
~.
Initialisation ~v , w
=w
~ get different colours if there is a colour c
Refinement Step ~v , w
~ have different
and an i ≤ k such that ~v and w
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 .
I
Stable colouring is reached after at most nk steps.
I
Stable colouring can be computed in time O(nk log n).
21
Remarks
Running time
k-WL on n-vertex graphs G .
I
Stable colouring is reached after at most nk steps.
I
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
I
3-WL identifies almost all regular graphs (Bollobás 1982)
22
The Strength of Weisfeiler-Lehman
I
3-WL identifies almost all regular graphs (Bollobás 1982)
I
There is a k such that k-WL identifies all interval graphs
(Laubner 2010)
22
The Strength of Weisfeiler-Lehman
I
3-WL identifies almost all regular graphs (Bollobás 1982)
I
There is a k such that k-WL identifies all interval graphs
(Laubner 2010)
I
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
I
3-WL identifies almost all regular graphs (Bollobás 1982)
I
There is a k such that k-WL identifies all interval graphs
(Laubner 2010)
I
For every class C of graphs with excluded minors there is a k
such that k-WL identifies all graphs in C (G. 2011)
I
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
bounded eigenvalue
multiplicities
excluded top. minors
excluded minors
bounded
degree
bounded
genus
bounded
colour classes
interval graphs
bounded
tree width
planar
trees
paths
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 ) ∈ {0, 1}V ×V , B = (bww 0 ) ∈ {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 ) ∈ {0, 1}V ×V , B = (bww 0 ) ∈ {0, 1}W ×W .
Observation
G and H are isomorphic iff there is a permutation matrix X such
that X T AX = B,
25
Integer Linear Program for GI
G , H n-vertex graphs with vertex sets V , W and adjacency
matrices A = (avv 0 ) ∈ {0, 1}V ×V , B = (bww 0 ) ∈ {0, 1}W ×W .
Observation
G and H are isomorphic iff there is a permutation matrix X such
that X T AX = 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 ) ∈ {0, 1}V ×V , B = (bww 0 ) ∈ {0, 1}W ×W .
Observation
G and H are isomorphic iff there is a permutation matrix X such
that X T AX = B, or equivalently
(?)
AX = XB.
ILP-Formulation
X
X
xvw 0 bw 0 w
avv 0 xv 0 w =
v 0 ∈V
n
X
w ∈W
w 0 ∈W
n
X
xvw =
xvw = 1
v ∈V
xvw ∈ {0, 1}
for all v ∈ V , w ∈ W .
25
LP-Relaxation
X
avv 0 xv 0 w =
v 0 ∈V
X
X
xvw 0 bw 0 w
w 0 ∈W
xvw =
w ∈W
xvw ≥ 0
X
xvw = 1
v ∈V
for all v ∈ V , w ∈ W .
26
LP-Relaxation
X
avv 0 xv 0 w =
v 0 ∈V
X
X
xvw 0 bw 0 w
w 0 ∈W
xvw =
w ∈W
X
xvw = 1
v ∈V
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.
X
X
avv 0 xp∪v 0 w =
xp∪vw 0 bw 0 w
v 0 ∈V
w 0 ∈W
27
Sherali-Adams Hierarchy
Level-k Sherali-Adams relaxation (Lk )
Variables xp for p ⊆ V × W with |p| ≤ k.
X
X
avv 0 xp∪v 0 w =
xp∪vw 0 bw 0 w
v 0 ∈V
w 0 ∈W
xX
∅ =1
X
xp∪vw =
xp∪vw = xp
w ∈W
for all |p| ≤ k − 1, v , w
v ∈V
27
Sherali-Adams Hierarchy
Level-k Sherali-Adams relaxation (Lk )
Variables xp for p ⊆ V × W with |p| ≤ k.
X
X
avv 0 xp∪v 0 w =
xp∪vw 0 bw 0 w
v 0 ∈V
w 0 ∈W
xX
∅ =1
X
xp∪vw =
xp∪vw = xp
w ∈W
xp ≥ 0
for all |p| ≤ k − 1, v , w
v ∈V
for all p
27
Sherali-Adams Hierarchy
Level-k Sherali-Adams relaxation (Lk )
Variables xp for p ⊆ V × W with |p| ≤ k.
X
X
avv 0 xp∪v 0 w =
xp∪vw 0 bw 0 w
v 0 ∈V
w 0 ∈W
xX
∅ =1
X
xp∪vw =
xp∪vw = xp
w ∈W
xp ≥ 0
for all |p| ≤ k − 1, v , w
v ∈V
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
X
v 0 ∈V
avv 0 xp∪v 0 w =
X
xp∪vw 0 bw 0 w
for all |p| ≤ k − 1, v , w
w 0 ∈W
29
An Exact Correspondence
COMPk
X
avv 0 xp∪v 0 w =
v 0 ∈V
CONTk
X
xp∪vw 0 bw 0 w
xX
∅ =1
X
xp∪vw =
xp∪vw = xp
w ∈W
for all |p| ≤ k − 1, v , w
w 0 ∈W
for all |p| ≤ k − 1, v , w
v ∈V
29
An Exact Correspondence
COMPk
X
avv 0 xp∪v 0 w =
v 0 ∈V
CONTk
xp∪vw 0 bw 0 w
Xp ≥ 0
for all |p| ≤ k − 1, v , w
w 0 ∈W
xX
∅ =1
X
xp∪vw =
xp∪vw = xp
w ∈W
NNk
X
for all |p| ≤ k − 1, v , w
v ∈V
for all |p| ≤ k
29
An Exact Correspondence
COMPk
X
avv 0 xp∪v 0 w =
v 0 ∈V
CONTk
xp∪vw 0 bw 0 w
for all |p| ≤ k − 1, v , w
w 0 ∈W
xX
∅ =1
X
xp∪vw =
xp∪vw = xp
w ∈W
NNk
X
for all |p| ≤ k − 1, v , w
v ∈V
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?
I
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?
I
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.
I
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.
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Is GI in solvable in polynomial time?
Why not?
I
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.
I
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.
I
There are complexity theoretic results (“hardness vs
derandomisation for AM”) indicating that at least GI is in
co-NP.
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