Locally Checkable Proofs Mika G o os & Jukka Suomela Helsinki - - PowerPoint PPT Presentation

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Jun 23, 2023 334 likes •733 views

Locally Checkable Proofs Mika G o os & Jukka Suomela Helsinki Institute for Information Technology HIIT G o os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 1 / 15 Basic Question 1 What global information can we infer

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SLIDE 1

Locally Checkable Proofs

Mika G¨

  • ¨
  • s & Jukka Suomela

Helsinki Institute for Information Technology HIIT

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 1 / 15

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SLIDE 2

Basic Question

1 What global information can we infer from local

structure?

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 2 / 15

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SLIDE 3

Basic Question

1 What global information can we infer from local

structure? . . .

2 Specifically: Can we prove to a distributed local

verifier that a graph has a certain global property?

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 2 / 15

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SLIDE 4

Local Algorithms

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 3 / 15

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SLIDE 5

Local Algorithms

15 34 92 65 43 84 27 77 31 30

Locality condition: constant running time t ∈ N

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 3 / 15

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SLIDE 6

Local Algorithms

Definition:

A : { } → {yes, no}

15 34 92 65 43 84 27 77 31 30

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 3 / 15

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SLIDE 7

Local Algorithms

yes no no no no no no no no no no no no no no no no no no no no yes yes yes yes yes yes yes yes yes yes

Graph is accepted

def

⇐ ⇒ all nodes output yes

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 3 / 15

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SLIDE 8

Locally Checkable Properties

[Naor & Stockmeyer, 1995]

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 4 / 15

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SLIDE 9

Locally Checkable Properties

e.g. Eulerian graphs

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 4 / 15

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SLIDE 10

Locally Checkable Properties

Graph Eulerian ⇐

⇒ all vertices have even degree

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 4 / 15

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SLIDE 11

Locally Checkable Proofs

1 Very few properties are locally checkable

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 5 / 15

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SLIDE 12

Locally Checkable Proofs

1 Very few properties are locally checkable 2 Extension: Add information to local neighbourhoods:

Proof labels:

P : V(G) → {0, 1}⋆

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 5 / 15

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SLIDE 13

Locally Checkable Proofs

1 Very few properties are locally checkable 2 Extension: Add information to local neighbourhoods:

Proof labels:

P : V(G) → {0, 1}⋆

3 “Proof Labelling Schemes”

[Korman, Kutten & Peleg, PODC 2005] [Korman & Kutten, 2007] [Fraigniaud, Korman & Peleg, 2010]

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 5 / 15

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SLIDE 14

Example: 3-Colourability

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 6 / 15

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SLIDE 15

Example: 3-Colourability

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 6 / 15

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SLIDE 16

Example: 3-Colourability

∃c : V → {1, 2, 3} s.t. all edges non-monochromatic

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 6 / 15

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SLIDE 17

Locally Checkable Proofs (LCP) — Definition

A graph property P admits locally checkable proofs of size f : N → N if there exists a local algorithm A so that

G ∈ P: There exists a proof P : V(G) → {0, 1} f (n(G))

so that A(G, P, v) outputs yes on all nodes v.

G /

∈ P: For every proof P, A(G, P, v) outputs no on

some node v.

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 7 / 15

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SLIDE 18

Complexity Theory Analogue

Locally checkable properties

≃

P

Locally checkable proofs

≃

NP

⇒

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 8 / 15

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SLIDE 19

Our Contributions

1 We study the Locally Checkable Proof (LCP) hierarchy

LCP(0) ⊂ LCP(O(1)) ⊂ LCP(O(log n)) ⊂ LCP(O(n2))

2 Extending the results of [Korman et al., 2005]

Our model is strictly stronger

3 Lower-bound constructions—using e.g.

Extremal graph theory Gadgets (from NP-completeness theory) Communication complexity

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 9 / 15

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SLIDE 20

Non-bipartiteness in LCP(O(log n))

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 10 / 15

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Non-bipartiteness in LCP(O(log n))

1 Find an odd cycle C

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  • ¨
  • s, Suomela (HIIT)

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SLIDE 22

Non-bipartiteness in LCP(O(log n))

L

1 Find an odd cycle C 2 Pick a leader L ∈ C

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 10 / 15

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SLIDE 23

Non-bipartiteness in LCP(O(log n))

L

1 2 3 4 5 6 7

1 Find an odd cycle C 2 Pick a leader L ∈ C 3 Equip C with node counters

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 10 / 15

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SLIDE 24

Non-bipartiteness in LCP(O(log n))

L

1 2 3 4 5 6 7

1 Find an odd cycle C 2 Pick a leader L ∈ C 3 Equip C with node counters 4 Prove the existence of a unique L using spanning tree methods

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 10 / 15

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Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 26

Proving Lower Bounds

1011 1 1 1 1 1 1 10 1101 1 11 1001 110 1 1 011 101 10

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 27

Proving Lower Bounds

  • dd
  • dd
  • dd
  • dd

even

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 28

Proving Lower Bounds yes yes yes yes no

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 29

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 30

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 31

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 32

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 33

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

slide-34
SLIDE 34

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 35

Proving Lower Bounds

1 Suppose non-bipartiteness admits proof of size o(log n)

with local algorithm A

2 Then A accepts odd cycles with short proofs:

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 11 / 15

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SLIDE 36

Local Proof Complexities 1

Class Proof size Graph property Graph family LCP(0) Eulerian graphs connected line graphs general LCP(O(1)) Θ(1) s–t reachability undirected Θ(1) s–t unreachability undirected Θ(1) s–t unreachability directed Θ(1) s–t connectivity = k planar Θ(1) bipartite graphs general Θ(1) even n(G) cycles LCP(O(log k)) O(log k) s–t connectivity = k general O(log k) chromatic number ≤ k general

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  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 12 / 15

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SLIDE 37

Local Proof Complexities 2

Class Proof size Graph property Graph family LCP(O(log n)) O(log n) any coLCP(0) property connected O(log n) any monadic Σ1

1 property

connected Θ(log n)

  • dd n(G)

cycles Θ(log n) chromatic number > 2 connected LCP(poly(n)) Θ(n) fixpoint-free symmetry trees Θ(n2) symmetric graphs connected Ω(n2/ log n) chromatic number > 3 connected O(n2) any computable property connected — — connected general

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  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 13 / 15

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Open Problems

1 The exact local proof complexity for many classical

problems remains unknown

2 Is it the case that, when ∆ = O(1),

LCP(O(1)) ⊆ NP ?

Note: we already know that

LCP(0) ⊆ P & LCP(O(log n))

  • NP

⊆ NP/poly

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 14 / 15

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SLIDE 39

Thank you!

G¨

  • ¨
  • s, Suomela (HIIT)

Locally Checkable Proofs 7th June 2011 15 / 15