Sunflowers: a bridge between complexity, combinatorics and - - PowerPoint PPT Presentation

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 = {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

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

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.

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˝

• s and Rado)

Let F be a w-set system such that |F| ≥ w! · 3w+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˝

• s and Rado)

Let F be a w-set system such that |F| ≥ w! · 3w+1, then it contains a sunflower of size 3. Notice that w! ≈ ww.

Jiapeng Zhang UCSD

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! · 3w+1, 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

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 Xp the p-biased distribution over X, where W ∼ Xp 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 Xp the p-biased distribution over X, where W ∼ Xp is sampled by including each x ∈ X in W independently with probability p. Notice that X1/2 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 Xp the p-biased distribution over X, where W ∼ Xp is sampled by including each x ∈ X in W independently with probability p. Notice that X1/2 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 Pr

W ∼Xp [∃S ∈ F : S ⊆ W ] > 1 − ε.

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∈FS. 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∈FS. 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
• f 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
• f a DNF is the maximal number of literals in a clause.

Example Let f = (x1 ∧ x2 ∧ x3) ∨ (x3 ∧ x4) ∨ (x1 ∧ x4 ∧ x5), 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
• f a DNF is the maximal number of literals in a clause.

Example Let f = (x1 ∧ x2 ∧ x3) ∨ (x3 ∧ x4) ∨ (x1 ∧ x4 ∧ x5), then f is a DNF. If x = 11100, then (x1 ∧ x2 ∧ x3) = 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 = {S1, . . . , Sm} ⇐ ⇒ fF(x) :=

• j∈[m]
• xi∈Sj

xi.

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 = {S1, . . . , Sm} ⇐ ⇒ fF(x) :=

• j∈[m]
• xi∈Sj

xi. Example Let F = {{x1, x2, x3}, {x3, x4}, {x1, x4, x5}}, 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 = {S1, . . . , Sm} ⇐ ⇒ fF(x) :=

• j∈[m]
• xi∈Sj

xi. Example Let F = {{x1, x2, x3}, {x3, x4}, {x1, x4, x5}}, then fF = (x1 ∧ x2 ∧ x3) ∨ (x3 ∧ x4) ∨ (x1 ∧ x4 ∧ x5).

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 Pr

x∼Xp [fF(x) = 1] ≥ 1 − ε

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Function approximation

Definition Let f : {0, 1}n → {0, 1} be a boolean function. We call a function g ε-approximates f if Pr

x∼{0,1}n[f (x) = g(x)] ≤ ε

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Function approximation

Definition Let f : {0, 1}n → {0, 1} be a boolean function. We call a function g ε-approximates f if Pr

x∼{0,1}n[f (x) = g(x)] ≤ ε

Lemma (Narrowing DNFs) Let f be a DNF of t clauses. Then it can be ε-approximated by a DNF of width at most log t + log(1/ε).

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

DNF sparsification 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) flower and fupper are ε-close. (iii) flower and fupper have size (w log(1/ε))O(w).

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

DNF sparsification 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) flower and f are ε-close. (iii) flower has size (1/ε)O(w).

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

DNF sparsification: upper bound approximation

Conjecture: Upper bound DNF sparsification Let f be a width-w DNF. Then for every ε > 0 there exists a width-w DNFs fupper such that (i) f (x) ≤ fupper(x) for all x. (ii) fupper and f are ε-close. (iii) fupper has size (1/ε)O(w).

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

From DNF sparsification to sunflower conjecture

A weak conjecture Let f be a monotone width-w DNF. Then for every ε > 0 there exists a width-w DNFs fupper such that (i) f (x) ≤ fupper(x) for all x. (ii) fupper and f are ε-close. (iii) fupper has size (log w/ε)O(w). Theorem (Lovett-Solomon-Zhang) 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 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|

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Regular set systems are satisfying

Theorem (Rossman, Li-Lovett-Zhang) Let F be a w-set system. Assume F is w-regular, then it is (1/2, 0.01)-satisfying.

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Regular set systems are satisfying

Theorem (Rossman, Li-Lovett-Zhang) Let F be a w-set system. Assume F is w-regular, then it is (1/2, 0.01)-satisfying. Conjecture Let F be a w-set system. Assume F is (log w)100-regular, then it is (1/2, 0.01)-satisfying.

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

From DNF compression to sunflowers

Conjecture: Upper bound DNF sparsification Let f be a monotone width-w DNF. Then for every ε > 0 there exists a width-w DNFs fupper such that (i) f (x) ≤ fupper(x) for all x. (ii) fupper and f are ε-close. (iii) fupper has size (log w/ε)O(w).

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

From DNF compression to sunflowers

Conjecture: Upper bound DNF sparsification Let f be a monotone width-w DNF. Then for every ε > 0 there exists a width-w DNFs fupper such that (i) f (x) ≤ fupper(x) for all x. (ii) fupper and f are ε-close. (iii) fupper has size (log w/ε)O(w). Theorem (Lovett-Solomon-Zhang) Assume the above conjecture holds. Then exists a constant c > 0, such that for any w-set system F, if it is (log w)c-regular, then it is also (1/2, 0.01)-satisfying.

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

From two disjoint sets to r 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 it is also (1/2, 0.01)-satisfying.

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

From two disjoint sets to r 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 it is also (1/2, 0.01)-satisfying. 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)r·c-regular, then it contains r pairwise disjoint sets.

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

A structure-vs-pseudorandomness trade-off

Structure v.s. Pseudorandomness Let F be a w-set system such that |F| ≥ (log w)r·c·w. If F is (log w)r·c, then it contains a r-sunflower. Otherwise, there is a set A such that |{S ∈ F : A ⊆ S}| ≥ |F| · (log w)−r·c·|A| ≥ (log w)−r·c·(w−|A|)

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

A structure-vs-pseudorandomness trade-off

Structure v.s. Pseudorandomness Let F be a w-set system such that |F| ≥ (log w)r·c·w. If F is (log w)r·c, then it contains a r-sunflower. Otherwise, there is a set A such that |{S ∈ F : A ⊆ S}| ≥ |F| · (log w)−r·c·|A| ≥ (log w)−r·c·(w−|A|) Notice that |{S \ A : S ∈ F ∧ A ⊆ S}| is a (w − |A|)-set system, then we can apply induction.

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Open problems

Question Could we confirm the upper bound DNF sparsification conjecture?

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Open problems

Question Could we confirm the upper bound DNF sparsification conjecture? Question Could we improve the sunflower bounds from lower bound DNF sparsification theorem?

Jiapeng Zhang UCSD

Sunflower families DNF sparsification Regular set systems Open problems

Thanks!

Jiapeng Zhang UCSD