Locally Checkable G aph P ope ies
Oli e Bach le and Tim Be gne
Depa men o Ma hema ics
TU Kaise slau e n
Fu u e Resea ch in Combina o ial Op imiza ion, 2021
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex.
⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
My kingdom
should be cubic.
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
My kingdom
should be cubic.
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
Looks good.
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
This is w ong,
no i y he king!
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
I also wan my
kingdom o be
bipa i e.
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
I also wan my
kingdom o be
bipa i e.
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
Is his eally
bipa i e?
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
Don’ o ge
you o de s!
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
Mo i a ion
In essence: wan o e i y ha a g aph has a ce ain p ope y.
Bu : we only see a local iew a ound e e y e ex. ⇒limi ed in o ma ion
O de has been
es ablished!
BA
B
A B
A
AB
A
B A
B
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 17
The Se ing
▶Gis a class o g aphs (all undi ec ed g aphs).
▶F⊆ G is a subse o Gsa is ying a ce ain p ope y (cubic,bipa i e).
▶Each e ex o a g aph has an iden i y, which a e
▶dis inc (de aul ), o
▶iden ical, in which case he g aph is anonymous.
▶Ve ices also ha e labels, which con ain p oblem-speci ic in o ma ion.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 4 / 17
The Se ing
▶Gis a class o g aphs (all undi ec ed g aphs).
▶F⊆ G is a subse o Gsa is ying a ce ain p ope y (cubic,bipa i e).
▶Each e ex o a g aph has an iden i y, which a e
▶dis inc (de aul ), o
▶iden ical, in which case he g aph is anonymous.
▶Ve ices also ha e labels, which con ain p oblem-speci ic in o ma ion.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 4 / 17
The Se ing
▶Gis a class o g aphs (all undi ec ed g aphs).
▶F⊆ G is a subse o Gsa is ying a ce ain p ope y (cubic,bipa i e).
▶Each e ex o a g aph has an iden i y, which a e
▶dis inc (de aul ), o
▶iden ical, in which case he g aph is anonymous.
▶Ve ices also ha e labels, which con ain p oblem-speci ic in o ma ion.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 4 / 17
P o e s
De ini ion (P o e )
▶Ap o e ( o F) is a unc ion P ha maps G∈ F o a p oo o G.
▶The size o Pis he maximum size o any p oo i assigns o F.
De ini ion (P oo )
▶Ap oo o a g aph Gis a unc ion P:V(G)→ {0,1}∗ ha assigns a
bina y ce i ica e o each e ex o G.
▶The size o a p oo is he leng h o i s longes ce i ica e.
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 5 / 17
P o e s
De ini ion (P o e )
▶Ap o e ( o F) is a unc ion P ha maps G∈ F o a p oo o G.
▶The size o Pis he maximum size o any p oo i assigns o F.
De ini ion (P oo )
▶Ap oo o a g aph Gis a unc ion P:V(G)→ {0,1}∗ ha assigns a
bina y ce i ica e o each e ex o G.
▶The size o a p oo is he leng h o i s longes ce i ica e.
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 5 / 17
P o e s
De ini ion (P o e )
▶Ap o e ( o F) is a unc ion P ha maps G∈ F o a p oo o G.
▶The size o Pis he maximum size o any p oo i assigns o F.
De ini ion (P oo )
▶Ap oo o a g aph Gis a unc ion P:V(G)→ {0,1}∗ ha assigns a
bina y ce i ica e o each e ex o G.
▶The size o a p oo is he leng h o i s longes ce i ica e.
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 5 / 17
P o e s
De ini ion (P o e )
▶Ap o e ( o F) is a unc ion P ha maps G∈ F o a p oo o G.
▶The size o Pis he maximum size o any p oo i assigns o F.
De ini ion (P oo )
▶Ap oo o a g aph Gis a unc ion P:V(G)→ {0,1}∗ ha assigns a
bina y ce i ica e o each e ex o G.
▶The size o a p oo is he leng h o i s longes ce i ica e.
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 5 / 17
P o e s
De ini ion (P o e )
▶Ap o e ( o F) is a unc ion P ha maps G∈ F o a p oo o G.
▶The size o Pis he maximum size o any p oo i assigns o F.
De ini ion (P oo )
▶Ap oo o a g aph Gis a unc ion P:V(G)→ {0,1}∗ ha assigns a
bina y ce i ica e o each e ex o G.
▶The size o a p oo is he leng h o i s longes ce i ica e.
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 5 / 17
Ve i ie s
De ini ion (Ve i ie )
▶A e i ie V( o G) is a unc ion ha maps iples (G,P, ) o {0,1}
and sa is ies
V(G,P, ) = V(G[N[ ]],P[N[ ]], ) o all G,P, .
▶Vaccep s a p oo Pa ∈V(G)i V(G,P, ) = 1.
▶Vaccep s a p oo P o a g aph Gi i accep s a all ∈V(G)and
▶V ejec s he p oo o he wise.
0
1
0 1
00
1 0
1
01
1
1
01
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 6 / 17
Ve i ie s
De ini ion (Ve i ie )
▶A e i ie V( o G) is a unc ion ha maps iples (G,P, ) o {0,1}
and sa is ies
V(G,P, ) = V(G[N[ ]],P[N[ ]], ) o all G,P, .
▶Vaccep s a p oo Pa ∈V(G)i V(G,P, ) = 1.
▶Vaccep s a p oo P o a g aph Gi i accep s a all ∈V(G)and
▶V ejec s he p oo o he wise.
0
1
0 1
00
1 0
1
01
1
1
01
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 6 / 17
Ve i ie s
De ini ion (Ve i ie )
▶A e i ie V( o G) is a unc ion ha maps iples (G,P, ) o {0,1}
and sa is ies
V(G,P, ) = V(G[N[ ]],P[N[ ]], ) o all G,P, .
▶Vaccep s a p oo Pa ∈V(G)i V(G,P, ) = 1.
▶Vaccep s a p oo P o a g aph Gi i accep s a all ∈V(G)and
▶V ejec s he p oo o he wise.
0
1
0 1
00
1 0
1
01
1
1
01
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 6 / 17
P oo Labelling Schemes
De ini ion (P oo Labelling Scheme)
▶A pai π= (P,V)is a p oo labelling scheme o F ⊆ G i
▶Vaccep s P(G) o all G∈ F.
▶V ejec s any g aph no in F.
▶The size o πis he size o i s p o e .
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 7 / 17
P oo Labelling Schemes
De ini ion (P oo Labelling Scheme)
▶A pai π= (P,V)is a p oo labelling scheme o F ⊆ G i
▶Vaccep s P(G) o all G∈ F.
▶V ejec s any g aph no in F.
▶The size o πis he size o i s p o e .
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 7 / 17
P oo Labelling Schemes
De ini ion (P oo Labelling Scheme)
▶A pai π= (P,V)is a p oo labelling scheme o F ⊆ G i
▶Vaccep s P(G) o all G∈ F.
▶V ejec s any g aph no in F.
▶The size o πis he size o i s p o e .
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 7 / 17
P oo Labelling Schemes
De ini ion (P oo Labelling Scheme)
▶A pai π= (P,V)is a p oo labelling scheme o F ⊆ G i
▶Vaccep s P(G) o all G∈ F.
▶V ejec s any g aph no in F.
▶The size o πis he size o i s p o e .
0
1
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 7 / 17
Examples
( aken om Göös, Suomela 2016)
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 8 / 17
Back o he Mo i a ion.
Example (Cubic G aphs)
▶No p oo needed: p o e assigns e e y e ex an emp y ce i ica e.
▶The e i ie a checks ha has deg ee 3.
▶Accep s exac ly he cubic g aphs.
⇒A p oo labelling scheme o size 0 exis s o cubic g aphs.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 9 / 17
Back o he Mo i a ion.
Example (Cubic G aphs)
▶No p oo needed: p o e assigns e e y e ex an emp y ce i ica e.
▶The e i ie a checks ha has deg ee 3.
▶Accep s exac ly he cubic g aphs.
⇒A p oo labelling scheme o size 0 exis s o cubic g aphs.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 9 / 17
Back o he Mo i a ion.
Example (Cubic G aphs)
▶No p oo needed: p o e assigns e e y e ex an emp y ce i ica e.
▶The e i ie a checks ha has deg ee 3.
▶Accep s exac ly he cubic g aphs.
⇒A p oo labelling scheme o size 0 exis s o cubic g aphs.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 9 / 17
Back o he Mo i a ion.
Example (Cubic G aphs)
▶No p oo needed: p o e assigns e e y e ex an emp y ce i ica e.
▶The e i ie a checks ha has deg ee 3.
▶Accep s exac ly he cubic g aphs.
⇒A p oo labelling scheme o size 0 exis s o cubic g aphs.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 9 / 17
Mo e Examples: Connec i i y
De ini ion (s -Connec i i y)
In he
di ec ed
s -connec i i y p oblem
▶Gis he class o all
di ec ed
g aphs wi h a e ex sand .
▶Fcon ains hose g aphs in which is eachable om s.
s s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 11 / 17
Mo e Examples: Connec i i y
De ini ion (s -Connec i i y)
In he
di ec ed
s -connec i i y p oblem
▶Gis he class o all
di ec ed
g aphs wi h a e ex sand .
▶Fcon ains hose g aphs in which is eachable om s.
s s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 11 / 17
Mo e Examples: Connec i i y
De ini ion (s -Connec i i y)
In he di ec ed s -connec i i y p oblem
▶Gis he class o all di ec ed g aphs wi h a e ex sand .
▶Fcon ains hose g aphs in which is eachable om s.
s s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 11 / 17
Mo e Examples: Connec i i y
Example (s -Connec i i y)
▶Use p oo o speci y a sho es pa h: p o e assigns ei he 0 o 1 o
each e ex o indica e whe he i is on a ixed sho es pa h.
▶The e i ie a checks ha has wo neighbou s wi h p oo 1 i i
has p oo 1 (o one i =so = ).
▶Accep s exac ly he g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size 1 exis s o s -connec i i y.
s
1
1
1 1
0 0 0
0
1
1
1
0 0
0
1
0
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 12 / 17
Mo e Examples: Connec i i y
Example (s -Connec i i y)
▶Use p oo o speci y a sho es pa h: p o e assigns ei he 0 o 1 o
each e ex o indica e whe he i is on a ixed sho es pa h.
▶The e i ie a checks ha has wo neighbou s wi h p oo 1 i i
has p oo 1 (o one i =so = ).
▶Accep s exac ly he g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size 1 exis s o s -connec i i y.
s
1
1
1 1
0 0 0
0
1
1
1
0 0
0
1
0
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 12 / 17
Mo e Examples: Connec i i y
Example (s -Connec i i y)
▶Use p oo o speci y a sho es pa h: p o e assigns ei he 0 o 1 o
each e ex o indica e whe he i is on a ixed sho es pa h.
▶The e i ie a checks ha has wo neighbou s wi h p oo 1 i i
has p oo 1 (o one i =so = ).
▶Accep s exac ly he g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size 1 exis s o s -connec i i y.
s
1
1
1 1
0 0 0
0
1
1
1
0 0
0
1
0
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 12 / 17
Mo e Examples: Connec i i y
Example (s -Connec i i y)
▶Use p oo o speci y a sho es pa h: p o e assigns ei he 0 o 1 o
each e ex o indica e whe he i is on a ixed sho es pa h.
▶The e i ie a checks ha has wo neighbou s wi h p oo 1 i i
has p oo 1 (o one i =so = ).
▶Accep s exac ly he g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size 1 exis s o s -connec i i y.
s
1
1
1 1
0 0 0
0
1
1
1
0 0
0
1
0
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 12 / 17
Mo e Examples: Connec i i y
Example (s -Connec i i y)
▶Use p oo o speci y a sho es pa h: p o e assigns ei he 0 o 1 o
each e ex o indica e whe he i is on a ixed sho es pa h.
▶The e i ie a checks ha has wo neighbou s wi h p oo 1 i i
has p oo 1 (o one i =so = ).
▶Accep s exac ly he g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size 1 exis s o s -connec i i y.
s
1
1
1 1
0 0 0
0
1
1
1
0 0
0
1
0
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 12 / 17
Mo e Examples: Connec i i y
Example (Di ec ed s -Connec i i y)
▶Use p oo o speci y a sho es pa h:
p o e assigns each e ex on a
ixed sho es pa h a poin e o i s successo .
▶The e i ie a checks ha i has a poin e , hen one o i s
p edecesso s poin s o i (unless =so = ).
▶Accep s exac ly he di ec ed g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size log(n)exis s o di . s -connec i i y.
s
1
1 1
1 1
0 0 0
1
1
1
s
ε ε
ε
εε
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 13 / 17
Mo e Examples: Connec i i y
Example (Di ec ed s -Connec i i y)
▶Use p oo o speci y a sho es pa h:
p o e assigns each e ex on a
ixed sho es pa h a poin e o i s successo .
▶The e i ie a checks ha i has a poin e , hen one o i s
p edecesso s poin s o i (unless =so = ).
▶Accep s exac ly he di ec ed g aphs whe e is eachable om s.
⇒A p oo labelling scheme o size log(n)exis s o di . s -connec i i y.
s
1
1 1
1 1
0 0 0
1
1
1
s
ε ε
ε
εε
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 13 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
Di ec ed is Ha de Than Undi ec ed Connec i i y
Goal: we wan o p o e ha
Theo em
No cons an -size p oo labelling scheme exis s o di ec ed s -connec i i y
on anonymous g aphs.
Idea:
s
u
s
u
ab
ab
ab
ab
ab
ab
u
u
ab
ab
u
u
Resul : o bids
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 15 / 17
P oo Ske ch
Goal: i e a i ely o bid mo e and mo e pai s
o bidden
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 16 / 17
P oo Ske ch
Wan : same colou ed o wa d-edges connec ed by back-edges
and o bidden
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 16 / 17
P oo Ske ch
How: use he Pigeon Hole P inciple
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 16 / 17
P oo Ske ch
Resul : pai o bidden in all g ey a eas
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 16 / 17
P oo Ske ch
Now: plug in ecu si ely o o bid mo e pai s
s
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 16 / 17
Summa y
We ha e:
▶ o mally de ined p oo labelling schemes,
▶illus a ed hese on se e al examples, and
▶showed ha small p oo s a e insu icien o di ec ed s -connec i i y.
Wha now?
▶Show ha small p oo s do no su ice o mo e powe ul e i ie s.
▶De e mine he op imal p oo sizes o o he p oblems.
Con ac : bach le @ma hema ik.uni-kl.de
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 17 / 17
Summa y
We ha e:
▶ o mally de ined p oo labelling schemes,
▶illus a ed hese on se e al examples, and
▶showed ha small p oo s a e insu icien o di ec ed s -connec i i y.
Wha now?
▶Show ha small p oo s do no su ice o mo e powe ul e i ie s.
▶De e mine he op imal p oo sizes o o he p oblems.
Con ac : bach le @ma hema ik.uni-kl.de
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 17 / 17
Re e ences
Lau en Feuilloley.
In oduc ion o local ce i ica ion, 2020.
Mika Göös and Jukka Suomela.
Locally checkable p oo s in dis ibu ed compu ing.
Theo y o Compu ing, 12(19):1–33, 2016.
A. Ko man, S. Ku en, and D. Peleg.
P oo labeling schemes.
Dis ibu ed Compu ing, 22:215–233, 2010.
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 1 / 2
Almos all O he Examples
Example (Uni e sal P oo Labelling Scheme)
▶Assume we can decide o a g aph G∈ G whe he G∈ F.
▶Use p oo o speci y he g aph: p o e assigns he adjacency ma ix A
and he co esponding ow o each e ex.
▶The e i ie a checks ha i s neighbou hood is co ec and ha
he g aph gi en by Ais in F.
▶Accep s exac ly he g aphs in F.
⇒The e exis s a uni e sal p oo labelling scheme o size O(n2).
A,7
A,1
A,8
A,2
A,9 A,3
A,10
A,4
A,11
A,5
A,12A,6
A,7
A,1
A,8
A,2
A,9 A,3
A,10
A,4
A,11
A,5
A,12A,6
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 2 / 2
Almos all O he Examples
Example (Uni e sal P oo Labelling Scheme)
▶Assume we can decide o a g aph G∈ G whe he G∈ F.
▶Use p oo o speci y he g aph: p o e assigns he adjacency ma ix A
and he co esponding ow o each e ex.
▶The e i ie a checks ha i s neighbou hood is co ec and ha
he g aph gi en by Ais in F.
▶Accep s exac ly he g aphs in F.
⇒The e exis s a uni e sal p oo labelling scheme o size O(n2).
A,7
A,1
A,8
A,2
A,9 A,3
A,10
A,4
A,11
A,5
A,12A,6
A,7
A,1
A,8
A,2
A,9 A,3
A,10
A,4
A,11
A,5
A,12A,6
O. Bach le and T. Be gne (TUK) Local Ve i ica ion FRICO 2021 2 / 2
