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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

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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

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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

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Sunflower families DNF sparsification Regular set systems Open problems

The Sunflower structure

Definition (Set systems) Let X be a set. A set system F = {S1, . . . , Sm} 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

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Sunflower families DNF sparsification Regular set systems Open problems

The Sunflower structure

Definition (Set systems) Let X be a set. A set system F = {S1, . . . , Sm} 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˝

s and Rado)

Given r sets S1, . . . , Sr ⊆ X where r ≥ 3. Denote as B = S1 ∩ · · · ∩ Sr We say it is a r-sunflower if for any i, j ∈ [r], Si ∩ Sj = B.

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

The Sunflower structure

Definition (Set systems) Let X be a set. A set system F = {S1, . . . , Sm} 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˝

s and Rado)

Given r sets S1, . . . , Sr ⊆ X where r ≥ 3. Denote as B = S1 ∩ · · · ∩ Sr We say it is a r-sunflower if for any i, j ∈ [r], Si ∩ Sj = B. We call B the kernal of this sunflower. In this talk, we focus on r = 3.

Jiapeng Zhang UCSD

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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

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Sunflower families DNF sparsification Regular set systems Open problems

The sunflower lemma (conjecture)

Lemma (Erd˝

s and Rado)

Let F be a w-set system such that |F| ≥ w! · 2w, then it contains a sunflower of size 3.

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

The sunflower lemma (conjecture)

Lemma (Erd˝

s and Rado)

Let F be a w-set system such that |F| ≥ w! · 2w, then it contains a sunflower of size 3. Notice that w! ≈ ww.

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

The sunflower lemma (conjecture)

Lemma (Erd˝

s and Rado)

Let F be a w-set system such that |F| ≥ w! · 2w, then it contains a sunflower of size 3. Notice that w! ≈ ww. Conjecture (Erd˝

s 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

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Sunflower families DNF sparsification Regular set systems Open problems

DNF sparsification

Jiapeng Zhang UCSD

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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 = (x1 ∧ x2) ∨ (x2 ∧ x3 ∧ x4) ∨ (x1 ∧ x4) ∨ (x2 ∧ x5) is a DNF of size 4 and width 3.

Jiapeng Zhang UCSD

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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 = {S1, . . . , Sm} ⇐ ⇒ fF(x) :=

j∈[m]
xi∈Sj

xi.

Jiapeng Zhang UCSD

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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 = {S1, . . . , Sm} ⇐ ⇒ fF(x) :=

j∈[m]
xi∈Sj

xi. {x1.x2}, {x2, x3, x4}, {x1, x4}, {x2, x5} ⇐ ⇒(x1 ∧ x2) ∨ (x2 ∧ x3 ∧ x4) ∨ (x1 ∧ x4) ∨ (x2 ∧ x5)

Jiapeng Zhang UCSD

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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, flower and fupper such that (i) flower(x) ≤ f (x) ≤ fupper(x) for all x. (ii) Pr[flower(x) = fupper(x)] ≤ ε for a uniform random x. (iii) flower and fupper have size (w log(1/ε))O(w).

Jiapeng Zhang UCSD

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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, flower and fupper such that (i) flower(x) ≤ f (x) ≤ fupper(x) for all x. (ii) Pr[flower(x) = fupper(x)] ≤ ε for a uniform random x. (iii) flower and fupper have size (w log(1/ε))O(w). Remark: The term ww comes Rossman’s approximate sunflower bound.

Jiapeng Zhang UCSD

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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 flower such that (i) flower(x) ≤ f (x) for all x. (ii) Pr[flower(x) = f (x)] ≤ ε for a uniform random x. (iii) flower has size (1/ε)O(w).

Jiapeng Zhang UCSD

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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 flower such that (i) flower(x) ≤ f (x) for all x. (ii) Pr[flower(x) = f (x)] ≤ ε for a uniform random x. (iii) flower has size (1/ε)O(w). Conjecture Could we find a fupper with the same bound as flower.

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

Regular set systems

Jiapeng Zhang UCSD

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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}| |F| ≤ κ−|A| That is, each variable is in 1/κ fraction sets, each pair in 1/κ2 fraction sets, and so on.

Jiapeng Zhang UCSD

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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}| |F| ≤ κ−|A| That is, each variable is in 1/κ fraction sets, each pair in 1/κ2 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

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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}| |F| ≤ κ−|A| That is, each variable is in 1/κ fraction sets, each pair in 1/κ2 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 FA := {S \ A : (S ∈ F) ∧ (A ⊆ S)}

Jiapeng Zhang UCSD

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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 fupper such that (i) f (x) ≤ fupper(x) for all x. (ii) Pr[fupper(x) = f (x)] ≤ ε for a uniform random x. (iii) fupper 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

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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, Prx[fF(x) = 1] ≥ 0.99]

Jiapeng Zhang UCSD

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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, Prx[fF(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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Sunflower families DNF sparsification Regular set systems Open problems

From two disjoint sets to r disjoint sets.

Lemma 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·r-regular, then F contains r disjoint sets.

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

From two disjoint sets to r disjoint sets.

Lemma 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·r-regular, then F contains r disjoint sets. Theorem Assume the upper bound DNF sparsification conjecture holds. For any w-set system F with |F| ≥ (log w)c·r·w, it contains a r-sunflower

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

Open problems

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

Open problems

Question Could we confirm the upper bound DNF sparsification conjecture?

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

Open problems

Question Could we confirm the upper bound DNF sparsification conjecture? Highlight conjecture There is a constant c. For any (log w)c-regular w-set system F, Pr

x [fF(x) = 1] ≥ 0.99

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

Open problems

Question Could we confirm the upper bound DNF sparsification conjecture? Highlight conjecture There is a constant c. For any (log w)c-regular w-set system F, Pr

x [fF(x) = 1] ≥ 0.99

I guess for each variable xi, the IfF(xi) ≪ 1/w.

Jiapeng Zhang UCSD

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Sunflower families DNF sparsification Regular set systems Open problems

Thanks!

Jiapeng Zhang UCSD