Sunflower families DNF sparsification Regular set systems Open problems From DNF compression to sunflower theorems via regularity Jiapeng Zhang UC San Diego → Harvard University July 20, 2019 Joint works with 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. In this talk, we focus on r = 3. 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 } . 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 ! · 2 w , 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 ! · 2 w , 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 ! · 2 w , 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 DNF sparsification 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 terms. The size of a DNF is the number of terms, and the width of a DNF is the maximal number of literals in a term. Example The function f = ( x 1 ∧ x 2 ) ∨ ( x 2 ∧ x 3 ∧ x 4 ) ∨ ( x 1 ∧ x 4 ) ∨ ( x 2 ∧ x 5 ) is a DNF of size 4 and width 3. 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 . j ∈ [ m ] x i ∈ S j 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 . j ∈ [ m ] x i ∈ S j { x 1 . x 2 } , { x 2 , x 3 , x 4 } , { x 1 , x 4 } , { x 2 , x 5 } ⇐ ⇒ ( x 1 ∧ x 2 ) ∨ ( x 2 ∧ x 3 ∧ x 4 ) ∨ ( x 1 ∧ x 4 ) ∨ ( x 2 ∧ x 5 ) Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems DNF compression from (approximate) sunflowers Theorem (Gopalan-Meka-Reingold) Let f be a width-w DNF. Then for every ε > 0 there exist two width-w DNFs, f lower and f upper such that (i) f lower ( x ) ≤ f ( x ) ≤ f upper ( x ) for all x. (ii) Pr[ f lower ( x ) � = f upper ( x )] ≤ ε for a uniform random x. (iii) f lower and f upper have size ( w log(1 /ε )) O ( w ) . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems DNF compression from (approximate) sunflowers Theorem (Gopalan-Meka-Reingold) Let f be a width-w DNF. Then for every ε > 0 there exist two width-w DNFs, f lower and f upper such that (i) f lower ( x ) ≤ f ( x ) ≤ f upper ( x ) for all x. (ii) Pr[ f lower ( x ) � = f upper ( x )] ≤ ε for a uniform random x. (iii) f lower and f upper have size ( w log(1 /ε )) O ( w ) . Remark: The term w w comes Rossman’s approximate sunflower bound. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems DNF compression beyond sunflowers Theorem (Lovett-Zhang) Let f be a width-w DNF. Then for every ε > 0 there exists a width-w DNFs f lower such that (i) f lower ( x ) ≤ f ( x ) for all x. (ii) Pr[ f lower ( x ) � = f ( x )] ≤ ε for a uniform random x. (iii) f lower has size (1 /ε ) O ( w ) . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems DNF compression beyond sunflowers Theorem (Lovett-Zhang) Let f be a width-w DNF. Then for every ε > 0 there exists a width-w DNFs f lower such that (i) f lower ( x ) ≤ f ( x ) for all x. (ii) Pr[ f lower ( x ) � = f ( x )] ≤ ε for a uniform random x. (iii) f lower has size (1 /ε ) O ( w ) . Conjecture Could we find a f upper with the same bound as f lower . Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Regular set systems Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Regular set systems Definition Let F be a set system on X , and κ > 0. We say that F is κ -regular if for every A ⊆ X , |{ S ∈ F : A ⊆ S }| ≤ κ −| A | |F| That is, each variable is in 1 /κ fraction sets, each pair in 1 /κ 2 fraction sets, and so on. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Regular set systems Definition Let F be a set system on X , and κ > 0. We say that F is κ -regular if for every A ⊆ X , |{ S ∈ F : A ⊆ S }| ≤ κ −| A | |F| That is, each variable is in 1 /κ fraction sets, each pair in 1 /κ 2 fraction sets, and so on. We want to find a minimum κ so that any κ -regular set system contains a three sunflower. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Regular set systems Definition Let F be a set system on X , and κ > 0. We say that F is κ -regular if for every A ⊆ X , |{ S ∈ F : A ⊆ S }| ≤ κ −| A | |F| That is, each variable is in 1 /κ fraction sets, each pair in 1 /κ 2 fraction sets, and so on. We want to find a minimum κ so that any κ -regular set system contains a three sunflower.If F is not κ -regular, then we can apply induction on F A := { S \ A : ( S ∈ F ) ∧ ( A ⊆ S ) } Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems From DNF sparsification to sunflowers Conjecture: Upper bound DNF sparsification Let f be a width- w monotone DNF. Then for every ε > 0 there exists a monotone width- w DNFs f upper such that (i) f ( x ) ≤ f upper ( x ) for all x . (ii) Pr[ f upper ( x ) � = f ( x )] ≤ ε for a uniform random x . (iii) f upper has size (1 /ε ) O ( w ) . Theorem (This work) Assume the above conjecture is true. There is a constant c > 1 such that, for any w-set system F with |F| ≥ (log w ) c · w , it contains a 3 -sunflower. Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Regular set system contains disjoint sets Theorem Assume the upper bound DNF sparsification conjecture holds. Then exists a constant c > 0 , such that for any w-set system F , if it is (log w ) c -regular, then Pr W [ ∃ S ∈ F , S ⊆ W ] ≥ 0 . 99 equivalently, Pr x [ f F ( x ) = 1] ≥ 0 . 99] Jiapeng Zhang UCSD
Sunflower families DNF sparsification Regular set systems Open problems Regular set system contains disjoint sets Theorem Assume the upper bound DNF sparsification conjecture holds. Then exists a constant c > 0 , such that for any w-set system F , if it is (log w ) c -regular, then Pr W [ ∃ S ∈ F , S ⊆ W ] ≥ 0 . 99 equivalently, Pr x [ f F ( x ) = 1] ≥ 0 . 99] Corollary Assume the above conjecture holds. Then for any (log w ) c -regular set system F , it contains two disjoint sets. Jiapeng Zhang UCSD
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