You say its only constant? Then something must be hyperfinite! And - PowerPoint PPT Presentation

You say it’s only constant? Then something must be hyperfinite! And I say your title is too long. Hendrik Fichtenberger July 20, 2019

Property Testing in a Nutshell planar 1

Property Testing in a Nutshell planar non-planar 1

Property Testing in a Nutshell planar non-planar non-planar 1

Property Testing in a Nutshell planar non-planar non-planar non-planar 1

Property Testing in a Nutshell planar non-planar non-planar non-planar time complexity: Ω(|𝑊|) 1

Property Testing in a Nutshell planar non-planar non-planar non-planar time complexity: Ω(|𝑊|) 1

Property Testing in a Nutshell planar non-planar non-planar non-planar time complexity: Ω(|𝑊|) very quite slightly 1

Property Testing in a Nutshell planar non-planar non-planar non-planar time complexity: Ω(|𝑊|) very quite slightly 1

Property Testing in a Nutshell planar non-planar non-planar non-planar 𝜗 1 0 # edge edits |𝑊|+|𝐹| 1

Property Testing in a Nutshell planar non-planar non-planar non-planar 3 3 𝜗 1 0 # edge edits |𝑊|+|𝐹| 1 accept w.p. > 2 reject w.p. > 2

Property Testing in a Nutshell 1 𝜗 -far 𝜗 -far 𝜗 -close |𝑊|+|𝐹| # edge edits 0 𝜗 planar 3 3 non-planar non-planar non-planar 1 accept w.p. > 2 reject w.p. > 2

Property Testing in a Nutshell 1 complexity: # queries to adjacency list entries 𝜗 -far 𝜗 -far 𝜗 -close |𝑊|+|𝐹| # edge edits 0 𝜗 planar 3 3 non-planar non-planar non-planar 1 accept w.p. > 2 reject w.p. > 2

Property Testing of Bounded Degree Graphs bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar 2

Property Testing of Bounded Degree Graphs bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... 2

Property Testing of Bounded Degree Graphs bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... bipartite, expander Θ(√𝑜) 2

Property Testing of Bounded Degree Graphs bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... bipartite, expander Θ(√𝑜) big picture? 2

Property Testing of Bounded Degree Graphs bounded degree graphs: ∀𝑤 ∈ 𝑊 ∶ 𝑒(𝑤) ≤ 𝑒, 𝑒 ∈ 𝑃(1) 𝑟(𝜗) planar, degree-regular, cycle-free, subgraph-free, connected, minor-free, hyperfinite, ... bipartite, expander Θ(√𝑜) big picture? bounded-degree graphs: litule known about constant-time testability 2

𝑙 -Disks and Frequency Vectors disk 𝑙 (𝑤) : subgraph induced by BFS (𝑤) of depth 𝑙 3

𝑙 -Disks and Frequency Vectors disk 1 ( ) disk 𝑙 (𝑤) : subgraph induced by BFS (𝑤) of depth 𝑙 3

𝑙 -Disks and Frequency Vectors disk 2 ( ) disk 1 ( ) disk 𝑙 (𝑤) : subgraph induced by BFS (𝑤) of depth 𝑙 3

𝑙 -Disks and Frequency Vectors disk 2 ( 1 ∑ ) ⋮ 0.6 0.4 freq 2 ( type calculate its share of vertices freq 𝑙 (𝐻) : for each 𝑙 -disk isomorphism by BFS (𝑤) of depth 𝑙 disk 𝑙 (𝑤) : subgraph induced ) disk 1 ( ) 3 ) = (

𝑙 -Disks and Frequency Vectors 0.4 locality feature frequency vector is a 1 ∑ ) ⋮ 0.6 3 disk 2 ( freq 2 ( type calculate its share of vertices freq 𝑙 (𝐻) : for each 𝑙 -disk isomorphism by BFS (𝑤) of depth 𝑙 disk 𝑙 (𝑤) : subgraph induced ) disk 1 ( ) ) = (

Constant-Qvery Testers accept Goldreich, Ron, STOC’09 1 2. accepts ifg ‖̃ freq Θ(𝑟) (𝐻) of freq Θ(𝑟) (𝐻) can be transformed into an algorithm that Every property tester with constant query complexity 𝑟 ∶= 𝑟(𝜗) Theorem [GR’09, …] Π Π Π Π Π 1 1 reject 4 1. computes an approximation ̃ Θ(𝑟) for any 𝐻 ′ ∈ Π freq Θ(𝑟) (𝐻) − freq Θ(𝑟) (𝐻 ′ )‖ 1 ≤

Hyperfinite Graphs Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5 In every planar graph, there exists a set of √𝑜 separators

Hyperfinite Graphs Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5 In every planar graph, there exists a set of √𝑜 separators

Hyperfinite Graphs 3 |𝑊| 3 |𝑊| Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5 In every planar graph, there exists a set of √𝑜 separators > 1 > 1

Hyperfinite Graphs remove 𝜗𝑜 edges components of size 𝜗 −2 Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5 In every planar graph, there exists a set of √𝑜 separators

Hyperfinite Graphs 𝜍(0.5) 𝜍(0.5) 𝜍(0.5) 𝜗 = 0.5 Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5

Hyperfinite Graphs 𝜗 = 0.5 𝜍(0.5) 𝜍(0.5) 𝜍(0.5) Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5

Hyperfinite Graphs 𝜗 = 0.5 𝜗 = 0.6 𝜍(0.6) 𝜍(0.6) 𝜍(0.6) Definition (𝝑, 𝐭) -hyperfinite: can remove at most 𝜗𝑒𝑜 edges to obtain connected components of size at most 𝑡 𝝇 -hyperfinite: (𝜗, 𝜍(𝜗)) -hyperfinite for all 𝜗 ∈ (0, 1] 5

The Story so Far Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍 -hyperfinite Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍 -hyperfinite [GR’09] Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍 -hyperfinite [GR’09] [NS’11] Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍 -hyperfinite [GR’09] [NS’11] Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍 -hyperfinite [GR’09] [NS’11] e.g. Π = connectivity contains expanders Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6

The Story so Far Π has constant query complexity Π is characterized by its k-disk distribution Π is 𝜍 -hyperfinite [GR’09] [NS’11] e.g. Π = connectivity contains expanders [FPS’19] Newman, Sohler, STOC’11; Fichtenberger, Peng, Sohler, SODA’19 6 ∃ Π ′ ⊆ Π : Π ′ is 𝜍 ′ -hyperfinite

Small Frequency-Preserver Graphs freq 𝑙 (𝐻) freq 𝑙 (𝐼) 𝜀 𝐻 𝐼 𝑜 𝑁(𝜀, 𝑙) Theorem [Alon’11] For every 𝜀, 𝑙 > 0 , there exists 𝑁(𝜀, 𝑙) such that for every 𝐻 there Alon, ’11, see: Lovász, Large Networks and Graph Limits , Proposition 19.10 7 exists 𝐼 of size at most 𝑁(𝜀, 𝑙) and ‖ freq 𝑙 (𝐻) − freq 𝑙 (𝐼)‖ 1 < 𝜀 .

Small Frequency-Preserver Graphs freq 𝑙 (𝐻) freq 𝑙 (𝐼) 𝜀 𝐻 𝐼 𝑜 𝑁(𝜀, 𝑙) For ρ-hyperfinite graphs M(δ,k) is at most Theorem [Alon’11] For every 𝜀, 𝑙 > 0 , there exists 𝑁(𝜀, 𝑙) such that for every 𝐻 there Alon, ’11, see: Lovász, Large Networks and Graph Limits , Proposition 19.10 7 exists 𝐼 of size at most 𝑁(𝜀, 𝑙) and ‖ freq 𝑙 (𝐻) − freq 𝑙 (𝐼)‖ 1 < 𝜀 .

Putuing Everything Into The Property Testing Blender in Π 𝑜 rel. 𝚸 size hypf. ? freq. v. change original start w/ 𝐻 ∈ Π 8

Putuing Everything Into The Property Testing Blender original < 𝜀/2 ? ? freq. pres. 𝑃(1) start w/ 𝐻 ∈ Π change in Π freq. v. ? hypf. size rel. 𝚸 𝑜 8

Putuing Everything Into The Property Testing Blender freq. pres. = 0 𝜗 -close to Π blow-up hyperfinite (0, 𝑁(𝜀, 𝑙)) - 𝑜 < 𝜀/2 ? ? 𝑃(1) in Π start w/ 𝐻 ∈ Π original change freq. v. ? hypf. size rel. 𝚸 𝑜 8