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

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 from local structure? G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 2 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 2 / 15

Local Algorithms G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 3 / 15

Local Algorithms 92 34 31 77 15 84 65 43 30 27 Locality condition: constant running time t ∈ N G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 3 / 15

Local Algorithms Definition: 92 34 77 31 A : { } → { yes , no } 15 84 65 43 30 27 G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 3 / 15

Local Algorithms no no no no no yes no no no yes no yes yes no no no no yes yes no yes yes yes no no no no yes no no yes def ⇐ ⇒ all nodes output yes Graph is accepted G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 3 / 15

Locally Checkable Properties [Naor & Stockmeyer, 1995] G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 4 / 15

Locally Checkable Properties e.g. Eulerian graphs G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 4 / 15

Locally Checkable Properties Graph Eulerian ⇐ ⇒ all vertices have even degree G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 4 / 15

Locally Checkable Proofs 1 Very few properties are locally checkable G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 5 / 15

Locally Checkable Proofs 1 Very few properties are locally checkable 2 Extension: Add information to local neighbourhoods: P : V ( G ) → { 0, 1 } ⋆ Proof labels: G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 5 / 15

Locally Checkable Proofs 1 Very few properties are locally checkable 2 Extension: Add information to local neighbourhoods: P : V ( G ) → { 0, 1 } ⋆ Proof labels: 3 “Proof Labelling Schemes” [Korman, Kutten & Peleg, PODC 2005] [Korman & Kutten, 2007] [Fraigniaud, Korman & Peleg, 2010] G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 5 / 15

Example: 3-Colourability G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 6 / 15

Example: 3-Colourability G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 6 / 15

Example: 3-Colourability ∃ c : V → { 1, 2, 3 } s.t. all edges non-monochromatic G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 6 / 15

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 . ∈ P : For every proof P , A ( G , P , v ) outputs no on G / some node v . G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 7 / 15

Complexity Theory Analogue Locally checkable proofs Locally checkable properties ≃ ≃ P NP ⇒ G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 8 / 15

Our Contributions 1 We study the Locally Checkable Proof ( LCP ) hierarchy LCP ( 0 ) ⊂ LCP ( O ( 1 )) ⊂ LCP ( O ( log n )) ⊂ LCP ( O ( n 2 )) 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 G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 9 / 15

Non-bipartiteness in LCP ( O ( log n )) G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 10 / 15

Non-bipartiteness in LCP ( O ( log n )) 1 Find an odd cycle C G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 10 / 15

Non-bipartiteness in LCP ( O ( log n )) L 1 Find an odd cycle C 2 Pick a leader L ∈ C G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 10 / 15

Non-bipartiteness in LCP ( O ( log n )) 2 3 L 1 4 7 5 6 1 Find an odd cycle C 2 Pick a leader L ∈ C 3 Equip C with node counters G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 10 / 15

Non-bipartiteness in LCP ( O ( log n )) 2 3 L 1 4 7 5 6 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 G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 10 / 15

Proving Lower Bounds 1 Suppose non-bipartiteness admits proof of size o ( log n ) with local algorithm A G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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: 1011 10 0 1 0 101 1 1 011 1 0 1 0 0 1 1 0 1 0 0 0 110 10 1001 1101 0 11 1 0 G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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: odd odd even odd odd G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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: yes yes no yes yes G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

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¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 11 / 15

Local Proof Complexities 1 Class Proof size Graph property Graph family LCP ( 0 ) 0 Eulerian graphs connected 0 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 G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 12 / 15

Local Proof Complexities 2 Class Proof size Graph property Graph family LCP ( O ( log n )) O ( log n ) any coLCP ( 0 ) property connected any monadic Σ 1 O ( log n ) 1 property connected Θ ( log n ) odd n ( G ) cycles Θ ( log n ) chromatic number > 2 connected LCP ( poly ( n )) Θ ( n ) fixpoint-free symmetry trees Θ ( n 2 ) symmetric graphs connected Ω ( n 2 / log n ) chromatic number > 3 connected O ( n 2 ) any computable property connected — — connected general G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 13 / 15

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 � � NP LCP ( 0 ) ⊆ P LCP ( O ( log n )) & ⊆ NP / poly G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 14 / 15

Thank you! G¨ o¨ os, Suomela (HIIT) Locally Checkable Proofs 7th June 2011 15 / 15