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 ≥ ǫ n 2 edges to make it 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 ≥ ǫ n 2 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 ≥ ǫ n 2 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) 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 ≥ ǫ n 2 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) edi’s graph regularity lemma. Gives C ( ǫ ) = 2 2 ... 2 Proof: By Szemer´ 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 3

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” 3

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) 3

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 ) = 2 2 ... 2 height poly(1 /ǫ ) Remark: C ( ǫ ) = poly(1 /ǫ ) works if G = F n p with p fixed [Fox–Lov´ asz ’17], but not for G = Z / N Z 3

Spoiler For testing bipartite patterns, poly(1 /ǫ ) samples suffice 4

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 5

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 < ǫ n 2 edges 5

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 < ǫ n 2 edges ◮ M ( ǫ ) = 2 2 2 ... 2 tower of height ǫ − O (1) (cannot be improved [Gowers]) ◮ Removal lemma holds with δ = M − O (1) = 1 / 2 2 2 ... 2 (possibly could be improved, but not beyond ǫ C log(1 /ǫ ) when H = K 3 ) 5

When can you guarantee poly(1 /ǫ ) bounds? 6

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] 6

What is VC dimension? Let S be a collection of subsets of Ω dim VC S := 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 ′ 7

What is VC dimension? Let S be a collection of subsets of Ω dim VC S := 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 R 2 is 3 7

What is VC dimension? Let S be a collection of subsets of Ω dim VC S := 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 R 2 is 3 VC dimension of a graph G is defined to be the VC dimension of the collection of vertex neighborhoods (Ω = V ( G )): dim VC G := dim VC { N ( v ) : v ∈ V ( G ) } 7

Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph 8

Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) 8

Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) 8

Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) don’t care Bipartite H as a bi-induced subgraph (similar to induced H G but don’t care about edges inside each bipartition) don’t care 8

Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) don’t care Bipartite H as a bi-induced subgraph (similar to induced H G but don’t care about edges inside each bipartition) don’t care dim VC G < d ⇐ ⇒ G forbids the following as a bi-induced subgraph: 1 00 1 000 0 1 001 1 01 2 010 011 10 3 2 100 101 11 110 111 d = 1 d = 2 d = 3 8

Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) don’t care Bipartite H as a bi-induced subgraph (similar to induced H G but don’t care about edges inside each bipartition) don’t care dim VC G < d ⇐ ⇒ G forbids the following as a bi-induced subgraph: 1 00 1 000 0 1 001 1 01 2 010 011 10 3 2 100 101 11 110 111 d = 1 d = 2 d = 3 Conversely, if G is bi-induced- H -free, then dim VC G = O H (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 2 k vertices. 9