Property testing for bipartite patterns Yufei Zhao (MIT) Joint work - - PowerPoint PPT Presentation

Property testing for bipartite patterns

Yufei Zhao (MIT) Joint work with Noga Alon (Princeton) and Jacob Fox (Stanford) June, 2018

1

Testing triangle-free-ness

[Rubinfield and Sudan ’96] [Goldreich, Goldwasser, Ron ’98] Goal: determine if an n-vertex graph is triangle-free or ǫ-far from triangle free ǫ-far from triangle-free: need to delete ≥ ǫn2 edges to make it triangle-free.

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Testing triangle-free-ness

[Rubinfield and Sudan ’96] [Goldreich, Goldwasser, Ron ’98] Goal: determine if an n-vertex graph is triangle-free or ǫ-far from triangle free ǫ-far from triangle-free: need to delete ≥ ǫn2 edges to make it triangle-free. Algorithm: Sample C(ǫ) triples at random

◮ If never see a triangle, then output “triangle-free” ◮ Else, output “ǫ-far from triangle-free”

2

Testing triangle-free-ness

[Rubinfield and Sudan ’96] [Goldreich, Goldwasser, Ron ’98] Goal: determine if an n-vertex graph is triangle-free or ǫ-far from triangle free ǫ-far from triangle-free: need to delete ≥ ǫn2 edges to make it triangle-free. Algorithm: Sample C(ǫ) triples at random

◮ If never see a triangle, then output “triangle-free” ◮ Else, output “ǫ-far from triangle-free”

Theorem (Ruzsa–Szemer´ edi 1976)

For every ǫ > 0 there exists a C(ǫ) > 0 so that this algorithm succeeds with probability > 2/3 (one-sided error)

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Testing triangle-free-ness

[Rubinfield and Sudan ’96] [Goldreich, Goldwasser, Ron ’98] Goal: determine if an n-vertex graph is triangle-free or ǫ-far from triangle free ǫ-far from triangle-free: need to delete ≥ ǫn2 edges to make it triangle-free. Algorithm: Sample C(ǫ) triples at random

◮ If never see a triangle, then output “triangle-free” ◮ Else, output “ǫ-far from triangle-free”

Theorem (Ruzsa–Szemer´ edi 1976)

For every ǫ > 0 there exists a C(ǫ) > 0 so that this algorithm succeeds with probability > 2/3 (one-sided error) Proof: By Szemer´ edi’s graph regularity lemma. Gives C(ǫ) = 22...2 height poly(1/ǫ) Remark: False with C(ǫ) = poly(1/ǫ)

2

Testing sum-free-ness

Goal: determine if A ⊂ G (abelian group) is sum-free or or ǫ-far from sum-free sum-free: no solutions to x + y = z ǫ-far from sum-free: need to remove ≥ ǫ|G| elements to make sum-free

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Testing sum-free-ness

Goal: determine if A ⊂ G (abelian group) is sum-free or or ǫ-far from sum-free sum-free: no solutions to x + y = z ǫ-far from sum-free: need to remove ≥ ǫ|G| elements to make sum-free Algorithm: Sample C(ǫ) triples (x, y, x + y) ∈ G 3 at random

◮ If never see x, y, x + y ∈ A, then output “sum-free” ◮ Else, output “ǫ-far from sum-free”

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Testing sum-free-ness

Goal: determine if A ⊂ G (abelian group) is sum-free or or ǫ-far from sum-free sum-free: no solutions to x + y = z ǫ-far from sum-free: need to remove ≥ ǫ|G| elements to make sum-free Algorithm: Sample C(ǫ) triples (x, y, x + y) ∈ G 3 at random

◮ If never see x, y, x + y ∈ A, then output “sum-free” ◮ Else, output “ǫ-far from sum-free”

Theorem (Green 2005)

For every ǫ > 0 there exists a C(ǫ) > 0 so that this algorithm succeeds with probability > 2/3 (one-sided error)

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Testing sum-free-ness

Goal: determine if A ⊂ G (abelian group) is sum-free or or ǫ-far from sum-free sum-free: no solutions to x + y = z ǫ-far from sum-free: need to remove ≥ ǫ|G| elements to make sum-free Algorithm: Sample C(ǫ) triples (x, y, x + y) ∈ G 3 at random

◮ If never see x, y, x + y ∈ A, then output “sum-free” ◮ Else, output “ǫ-far from sum-free”

Theorem (Green 2005)

For every ǫ > 0 there exists a C(ǫ) > 0 so that this algorithm succeeds with probability > 2/3 (one-sided error) Proof: By regularity lemma. Gives C(ǫ) = 22...2 height poly(1/ǫ) Remark: C(ǫ) = poly(1/ǫ) works if G = Fn

p with p fixed [Fox–Lov´

asz ’17], but not for G = Z/NZ

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Spoiler

For testing bipartite patterns, poly(1/ǫ) samples suffice

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Szemer´ edi’s graph regularity lemma

Graph regularity lemma

For every ǫ > 0 there exists M = M(ǫ) so that every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ-regular

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Szemer´ edi’s graph regularity lemma

Graph regularity lemma

For every ǫ > 0 there exists M = M(ǫ) so that every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ-regular

Graph removal lemma

For every ∀ǫ > 0 and graph H there is some δ = δ(H, ǫ) > 0 so that every n-vertex graph with H-density < δ can be made H-free by removing < ǫn2 edges

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Szemer´ edi’s graph regularity lemma

Graph regularity lemma

For every ǫ > 0 there exists M = M(ǫ) so that every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ-regular

Graph removal lemma

For every ∀ǫ > 0 and graph H there is some δ = δ(H, ǫ) > 0 so that every n-vertex graph with H-density < δ can be made H-free by removing < ǫn2 edges

◮ M(ǫ) = 222...2

tower of height ǫ−O(1) (cannot be improved [Gowers])

◮ Removal lemma holds with δ = M−O(1) = 1/222...2

(possibly could be improved, but not beyond ǫC log(1/ǫ) when H = K3)

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When can you guarantee poly(1/ǫ) bounds?

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When can you guarantee poly(1/ǫ) bounds?

For a graph with bounded VC dimension:

◮ Vertices can be partitioned into ǫ−O(1) parts ◮ All but ǫ-fraction of pairs of vertex parts have densities ≤ ǫ or ≥ 1 − ǫ

[Alon–Fischer–Newman, Lov´ asz–Szegedy]

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What is VC dimension?

Let S be a collection of subsets of Ω dimVCS := size of the largest shattered subset of Ω U ⊂ Ω is shattered if for every U′ ⊆ U there exists T ∈ S such that T ∩ U = U′

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What is VC dimension?

Let S be a collection of subsets of Ω dimVCS := size of the largest shattered subset of Ω U ⊂ Ω is shattered if for every U′ ⊆ U there exists T ∈ S such that T ∩ U = U′ E.g., the VC-dimension of the collection of half-planes in R2 is 3

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What is VC dimension?

Let S be a collection of subsets of Ω dimVCS := size of the largest shattered subset of Ω U ⊂ Ω is shattered if for every U′ ⊆ U there exists T ∈ S such that T ∩ U = U′ E.g., the VC-dimension of the collection of half-planes in R2 is 3 VC dimension of a graph G is defined to be the VC dimension of the collection of vertex neighborhoods (Ω = V (G)): dimVCG := dimVC{N(v) : v ∈ V (G)}

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Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph

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Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph

H as a subgraph of G (all edges of H are present in G)

H G

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Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph

H as a subgraph of G (all edges of H are present in G)

H G

H as an induced subgraph of G (all edges of H are present in G and non-edges are not present)

H G

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Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph

H as a subgraph of G (all edges of H are present in G)

H G

H as an induced subgraph of G (all edges of H are present in G and non-edges are not present)

H G

Bipartite H as a bi-induced subgraph (similar to induced but don’t care about edges inside each bipartition)

don’t care don’t care H G

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Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph

H as a subgraph of G (all edges of H are present in G)

H G

H as an induced subgraph of G (all edges of H are present in G and non-edges are not present)

H G

Bipartite H as a bi-induced subgraph (similar to induced but don’t care about edges inside each bipartition)

don’t care don’t care H G

dimVCG < d ⇐ ⇒ G forbids the following as a bi-induced subgraph:

1 1 1 2 00 01 10 11 1 2 3 000 001 010 011 100 101 110 111 d = 1 d = 2 d = 3

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Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph

H as a subgraph of G (all edges of H are present in G)

H G

H as an induced subgraph of G (all edges of H are present in G and non-edges are not present)

H G

Bipartite H as a bi-induced subgraph (similar to induced but don’t care about edges inside each bipartition)

don’t care don’t care H G

dimVCG < d ⇐ ⇒ G forbids the following as a bi-induced subgraph:

1 1 1 2 00 01 10 11 1 2 3 000 001 010 011 100 101 110 111 d = 1 d = 2 d = 3

Conversely, if G is bi-induced-H-free, then dimVCG = OH(1)

8

When can you guarantee poly(1/ǫ) bounds?

Regularity lemma for graphs of bounded VC dimension

For a fixed bipartite H, if G is bi-induced-H-free, then G has a vertex partition into ǫ−O(1) parts so that all but ≤ ǫ-fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ.

9

When can you guarantee poly(1/ǫ) bounds?

Regularity lemma for graphs of bounded VC dimension

For a fixed bipartite H, if G is bi-induced-H-free, then G has a vertex partition into ǫ−O(1) parts so that all but ≤ ǫ-fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ. A graph is k-stable if it does not contain a bi-induced half-graph on 2k vertices.

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When can you guarantee poly(1/ǫ) bounds?

Regularity lemma for graphs of bounded VC dimension

For a fixed bipartite H, if G is bi-induced-H-free, then G has a vertex partition into ǫ−O(1) parts so that all but ≤ ǫ-fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ. A graph is k-stable if it does not contain a bi-induced half-graph on 2k vertices.

Stable regularity lemma [Malliaris–Shelah]

If the graph is k-stable, then we can furthermore guarantee that every pair of parts has density ≤ ǫ or ≥ 1 − ǫ.

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Arithmetic setting

G abelian group, A ⊂ G dimVCA := dimVC{A + x : x ∈ G} = dimVCCayleyGraph(G, A)

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Arithmetic setting

G abelian group, A ⊂ G dimVCA := dimVC{A + x : x ∈ G} = dimVCCayleyGraph(G, A) We say that A contains a bi-induced copy of a bipartite graph H = (U ∪ V , E) if the same is true for CayleyGraph(G, A), or equivalently, there exists x1, . . . , x|U|, y1, . . . , y|V | ∈ G such that ∀(i, j) ∈ U × V : xi + yj ∈ A if and only if (i, j) ∈ E

x1 x2 x3 x4 x5 y1 y2 y3 y4 y5

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Regularity lemmas with constraints

Graph regularity:

◮ Bounded VC-dimension (i.e., forbidding a bi-induced subgraph)

[Alon–Fischer–Newman]: vertex-partition into ≤ ǫ−O(1) parts so that all but ≤ ǫ-fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ

◮ Stability (i.e., forbidding a fixed-size half-graph) [Malliaris–Shelah]: furthermore

every pair of parts has density ≤ ǫ or ≥ 1 − ǫ

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Regularity lemmas with constraints

Graph regularity:

◮ Bounded VC-dimension (i.e., forbidding a bi-induced subgraph)

[Alon–Fischer–Newman]: vertex-partition into ≤ ǫ−O(1) parts so that all but ≤ ǫ-fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ

◮ Stability (i.e., forbidding a fixed-size half-graph) [Malliaris–Shelah]: furthermore

every pair of parts has density ≤ ǫ or ≥ 1 − ǫ Arithmetic regularity: A ⊂ G = Fn

p, p fixed

(general groups: Conant–Pillay–Terry, Terry–Wolf)

◮ Stability [Terry–Wolf]: there exists a subspace H ≤ G with [G : H] ≤ eǫ−O(1) such

that for all x ∈ G, |A ∩ (H + x)| ≤ ǫ|H| or ≥ (1 − ǫ)|H| (†)

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Regularity lemmas with constraints

Graph regularity:

◮ Bounded VC-dimension (i.e., forbidding a bi-induced subgraph)

[Alon–Fischer–Newman]: vertex-partition into ≤ ǫ−O(1) parts so that all but ≤ ǫ-fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ

◮ Stability (i.e., forbidding a fixed-size half-graph) [Malliaris–Shelah]: furthermore

every pair of parts has density ≤ ǫ or ≥ 1 − ǫ Arithmetic regularity: A ⊂ G = Fn

p, p fixed

(general groups: Conant–Pillay–Terry, Terry–Wolf)

◮ Stability [Terry–Wolf]: there exists a subspace H ≤ G with [G : H] ≤ eǫ−O(1) such

that for all x ∈ G, |A ∩ (H + x)| ≤ ǫ|H| or ≥ (1 − ǫ)|H| (†)

◮ Bounded VC-dimension [Alon–Fox–Z.]: there exists a subspace H ≤ G with

[G : H] ≤ ǫ−O(1) such that (†) holds for all but an ≤ ǫ-fraction of x ∈ G

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Applications to removal lemma and property testing

Recall the graph removal lemma: ∀ǫ∃δ: if an n-vertex graph has < δn3 triangles, and it can be made triangle-free by removing < ǫn2 edges.

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Applications to removal lemma and property testing

Recall the graph removal lemma: ∀ǫ∃δ: if an n-vertex graph has < δn3 triangles, and it can be made triangle-free by removing < ǫn2 edges.

Arithmetic removal lemma for bi-induced patterns [Alon–Fox–Z.]

Fix p and bipartite graph F. For every ǫ > 0, there exists δ = ǫO(|V (F)|3) such that if the bi-induced-F-density in A ⊂ Fn

p is < δ, then A can be made bi-induced-F-free by

adding/deleting < ǫpn elements.

Corollary: property testing for arithmetic bi-induced patterns

Using poly(1/ǫ) samples, one can distinguish, with probability > 2/3, subsets that are bi-induced-F-free from those that are ǫ-far from bi-induced-F-free. More generally holds for abelian groups with bounded exponent

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Open questions

◮ Bounds for Conant–Pillay–Terry regularity lemmas for general groups with

stability/bounded VC dimension hypotheses? (Proved via model theory. No bounds known. Maybe ǫ−O(1)?)

◮ Quantitative bounds for stability in abelian groups recently proved by Terry–Wolf 13

Open questions

◮ Bounds for Conant–Pillay–Terry regularity lemmas for general groups with

stability/bounded VC dimension hypotheses? (Proved via model theory. No bounds known. Maybe ǫ−O(1)?)

◮ Quantitative bounds for stability in abelian groups recently proved by Terry–Wolf

◮ Property testing for bi-induced arithmetic patterns in a general (abelian) group?

(No theorem known. Maybe ǫ−O(1) bounds?)

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Open questions

◮ Bounds for Conant–Pillay–Terry regularity lemmas for general groups with

stability/bounded VC dimension hypotheses? (Proved via model theory. No bounds known. Maybe ǫ−O(1)?)

◮ Quantitative bounds for stability in abelian groups recently proved by Terry–Wolf

◮ Property testing for bi-induced arithmetic patterns in a general (abelian) group?

(No theorem known. Maybe ǫ−O(1) bounds?)

◮ Property testing for induced arithmetic patterns?

(No general theorem known, even in Fn

p)

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