Sunflower families DNF sparsification Regular set systems Open problems Sunflowers: a bridge between complexity, combinatorics and pseudorandomness Jiapeng Zhang University of California, San Diego March 28, 2019 Joint works with Xin Li, Shachar Lovett and Noam Solomon Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Overview 1 Sunflower families 2 DNF sparsification 3 Regular set systems 4 Open problems Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems The Sunflower structure Definition (Set systems) Let X be a set. A set system F = { S 1 , . . . , S m } on X is a collection of subsets of X . We call F a w -set system if each set of F has size at most w . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems The Sunflower structure Definition (Set systems) Let X be a set. A set system F = { S 1 , . . . , S m } on X is a collection of subsets of X . We call F a w -set system if each set of F has size at most w . Definition (Sunflower, Erd˝ os and Rado) Given r sets S 1 , . . . , S r ⊆ X where r ≥ 3. Denote as B = S 1 ∩ · · · ∩ S r We say it is a r-sunflower if for any i , j ∈ [ r ], S i ∩ S j = B . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems The Sunflower structure Definition (Set systems) Let X be a set. A set system F = { S 1 , . . . , S m } on X is a collection of subsets of X . We call F a w -set system if each set of F has size at most w . Definition (Sunflower, Erd˝ os and Rado) Given r sets S 1 , . . . , S r ⊆ X where r ≥ 3. Denote as B = S 1 ∩ · · · ∩ S r We say it is a r-sunflower if for any i , j ∈ [ r ], S i ∩ S j = B . We call B the kernal of this sunflower. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Sunflowers: an example An example Let F = {{ a , b , x } , { a , b , y } , { a , b , z }} . Then F is a 3-sunflower. Its kernal is { a , b } , and petals are { x } , { y } , { z } . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Sunflowers: an example An example Let F = {{ a , b , x } , { a , b , y } , { a , b , z }} . Then F is a 3-sunflower. Its kernal is { a , b } , and petals are { x } , { y } , { z } . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems The sunflower lemma (conjecture) Lemma (Erd˝ os and Rado) Let F be a w-set system such that |F| ≥ w ! · 3 w +1 , then it contains a sunflower of size 3 . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems The sunflower lemma (conjecture) Lemma (Erd˝ os and Rado) Let F be a w-set system such that |F| ≥ w ! · 3 w +1 , then it contains a sunflower of size 3 . Notice that w ! ≈ w w . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems The sunflower lemma (conjecture) Lemma (Erd˝ os and Rado) Let F be a w-set system such that |F| ≥ w ! · 3 w +1 , then it contains a sunflower of size 3 . Notice that w ! ≈ w w . Conjecture (Erd˝ os and Rado) There is a constant C > 0 such that for any w -set system F with |F| ≥ C w , it contains a sunflower of size 3. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Satisfying probability Definition Let F a set system, we define its satisfying probability (under uniform distribution) as Pr W [ ∃ S ∈ F : S ⊆ W ] where W is a random subset of X . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Satisfying probability Definition Let F a set system, we define its satisfying probability (under uniform distribution) as Pr W [ ∃ S ∈ F : S ⊆ W ] where W is a random subset of X . Example Let F = {{ a , b , x } , { a , b , y } , { a , b , z }} , then its satisfying probability is 1 4 · 7 8 . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Satisfying set systems Definition ( p -biased distribution) Given a set X and 0 < p < 1, we denote by X p the p-biased distribution over X , where W ∼ X p is sampled by including each x ∈ X in W independently with probability p . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Satisfying set systems Definition ( p -biased distribution) Given a set X and 0 < p < 1, we denote by X p the p-biased distribution over X , where W ∼ X p is sampled by including each x ∈ X in W independently with probability p . Notice that X 1 / 2 is the uniform distribution. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Satisfying set systems Definition ( p -biased distribution) Given a set X and 0 < p < 1, we denote by X p the p-biased distribution over X , where W ∼ X p is sampled by including each x ∈ X in W independently with probability p . Notice that X 1 / 2 is the uniform distribution. Definition (Satisfying set system) Let F be a set system and let 0 < p , ε < 1. We say that F is ( p , ε )-satisfying if W ∼ X p [ ∃ S ∈ F : S ⊆ W ] > 1 − ε. Pr Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems A satisfying set system contains disjoint sets Lemma Let r ≥ 2 and F ⊆ P ( X ) be a (1 / r , 1 / r ) -satisfying set system. Then F contains r pairwise disjoint sets. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Approximate-sunflower Definition (Approximate sunflower, Rossman) Let F be a set system and let 0 < p , ε < 1. Denote B = ∩ S ∈F S . We call F is a ( p , ε )-approximate sunflower if the set system F ′ := { S \ B : S ∈ F} is ( p , ε )-satisfying. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Approximate-sunflower Definition (Approximate sunflower, Rossman) Let F be a set system and let 0 < p , ε < 1. Denote B = ∩ S ∈F S . We call F is a ( p , ε )-approximate sunflower if the set system F ′ := { S \ B : S ∈ F} is ( p , ε )-satisfying. Theorem (Rossman) Let F be a w-set system and let ε > 0 . If |F| ≥ w ! · (1 . 71 log(1 /ε ) / p ) w then F contains a ( p , ε ) -approximate sunflower. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Approximate sunflowers implies sunflowers Corollary Let F be a set system that contains a (1 / r , 1 / r ) -approximate sunflower, then it contains a r-sunflower. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Disjunctive Normal Forms Definition A DNF (Disjunctive Normal Form) is disjunction of conjunctive clauses. The size of a DNF is the number of clauses, and the width of a DNF is the maximal number of literals in a clause. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Disjunctive Normal Forms Definition A DNF (Disjunctive Normal Form) is disjunction of conjunctive clauses. The size of a DNF is the number of clauses, and the width of a DNF is the maximal number of literals in a clause. Example Let f = ( x 1 ∧ x 2 ∧ x 3 ) ∨ ( x 3 ∧ x 4 ) ∨ ( x 1 ∧ x 4 ∧ x 5 ) , then f is a DNF. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Disjunctive Normal Forms Definition A DNF (Disjunctive Normal Form) is disjunction of conjunctive clauses. The size of a DNF is the number of clauses, and the width of a DNF is the maximal number of literals in a clause. Example Let f = ( x 1 ∧ x 2 ∧ x 3 ) ∨ ( x 3 ∧ x 4 ) ∨ ( x 1 ∧ x 4 ∧ x 5 ) , then f is a DNF. If x = 11100, then ( x 1 ∧ x 2 ∧ x 3 ) = 1, thus f ( x ) = 1. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Set systems and monotone DNFs A DNF is monotone if it contains no negated variables. Monotone DNFs are in one-to-one correspondence with set systems. Formally, � � F = { S 1 , . . . , S m } ⇐ ⇒ f F ( x ) := x i . x i ∈ S j j ∈ [ m ] Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Set systems and monotone DNFs A DNF is monotone if it contains no negated variables. Monotone DNFs are in one-to-one correspondence with set systems. Formally, � � F = { S 1 , . . . , S m } ⇐ ⇒ f F ( x ) := x i . x i ∈ S j j ∈ [ m ] Example Let F = {{ x 1 , x 2 , x 3 } , { x 3 , x 4 } , { x 1 , x 4 , x 5 }} , then Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Set systems and monotone DNFs A DNF is monotone if it contains no negated variables. Monotone DNFs are in one-to-one correspondence with set systems. Formally, � � F = { S 1 , . . . , S m } ⇐ ⇒ f F ( x ) := x i . x i ∈ S j j ∈ [ m ] Example Let F = {{ x 1 , x 2 , x 3 } , { x 3 , x 4 } , { x 1 , x 4 , x 5 }} , then f F = ( x 1 ∧ x 2 ∧ x 3 ) ∨ ( x 3 ∧ x 4 ) ∨ ( x 1 ∧ x 4 ∧ x 5 ) . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Satisfying DNFs Lemma A set system F is ( p , ε ) -satisfying if and only if x ∼ X p [ f F ( x ) = 1] ≥ 1 − ε Pr Jiapeng Zhang UCSD
Recommend
More recommend
