Approximate Sunflowers

Transcription

1 Approximate Sunflowers Benamin Rossman January 26, 2019 Abstract A (p, ε-approximate sunflower is a family of sets S with the property that a p-random subset of the universe is 1 ε likely to contain a set of the form A \ I where A S and I is the intersection of all elements of S. In this note, we give a proof of the Approximate Sunflower Theorem from [Ros14] (with a slightly sharper bound showing that every l-uniform set system of size l!((t+ 1 2 /pl contains a (p, e t -approximate sunflower. This result was originally applied to obtain monotone circuit lower bounds for the clique problem on Erdős-Rényi random graphs. The Approximate Sunflower Theorem has subsequently found applications in the sparsification of DNF formulas [GMR13] and was recently connected to questions on randomness extractors [LLZ18]. It has also been noted that improving the bound to f(p, t l for any function f(p, t (which does not depend on l would prove the notorious Sunflower Conecture [LZ18, LSZ18]. Throughout this note, let t > 0 and p (0, 1 be arbitrary real numbers, let l be a positive integer, and let be an arbitrary set. Let ( l denote the set of l-element subsets of, and let ( <l l 1 ( 0. We say that X is a p-random subset of, written X p, if X contains each element of independently with probability p. A set system over is a family S of subsets of. For B, let S B denote the set system S B : {A \ B : B A S}. Borrowing terminology from the literature on sunflowers, we define the core of S as the intersection C A S A of all elements of S; elements of S C are called petals of S. A set system S is a sunflower if its petals are pairwise disoint (equivalently: if all pairs of distinct elements in S have the same intersection. A set system S is l-uniform if A l for all A S (i.e., S ( l. The Erdős-Rado Sunflower Theorem [ER60] establishes that every sufficiently large l-uniform set system contains a sunflower of size k. Theorem 1 (Sunflower Theorem. Every l-uniform set system of size > l!(k 1 l contains a sunflower of size k. The following notion of approximate sunflowers was introduced in [Ros14]. (This was originally called quasi-sunflower. The much better name approximate sunflowers was suggested by Lovett and Zhang. Definition 2. A set system S over is a (p, ε-approximate sunflower if a p-random subset of contains a petal of S with probability > 1 ε. 1

2 Note that S contains a (p, ε-approximate sunflower if, and only if, there exists a set B such that P X p [ ( A S B A X ] > 1 ε. In [Ros14] I showed that every l-uniform set system of size l!(2.5t/p l contains a (p, e t -approximate sunflower. This note proves a slightly stronger bound by a more careful analysis of the argument in [Ros14]. Theorem 3 (Approximate Sunflower Theorem. Every l-uniform set system of size l!((t+ 1 2 /pl contains a (p, e t -approximate sunflower. The proof of Theorem 3 is by induction on l, similar to the proof of the Sunflower Theorem. A key tool in the argument is Janson s Inequality (Theorem 6. As we explain in Remark 10, the bound in Theorem 3 is essentially best possible by this method: obtaining a bound better than l!(t/p l (or any bound of the form f(p, t l without the l! factor appears to require a substantially different proof technique. An approach via randomness extractors was recently suggested by Li, Lovett and Zhang [LLZ18], who give an extractor-based proof of a quantitatively weaker version of Theorem 3 with the bound 2 2l ((l + 1.5t/p cl for a constant c > 1. Before presenting the proof of Theorem 3, we remark on the relationship between sunflowers and approximate sunflowers. Proposition 4 (Sunflower Approximate Sunflower. Every sunflower S of size k is a (p, e kpl - approximate sunflower where l is the size of largest petal in S. Proof. Let S be an l-uniform sunflower over with petals A 1,..., A k. For X p, we have P[ X contains no petal of S ] k i1 P[ A i X ] (1 p l k e kpl. A cute relationship in the other direction was communicated to me by Jiapeng Zhang (an unpublished observation of Lovett, Solomon and Zhang [LSZ18]. Proposition 5 (Approximate Sunflower Sunflower. Every ( 1 k, 1 k -approximate sunflower contains a sunflower of size k. Proof. Let S be a ( 1 k, 1 k -approximate sunflower. Let X 1 X k be a uniform random partition of. Note that each X i individually is a 1 k -random subset of. Let I i {0, 1} be the indicator 1[ X i contains a petal of S ]. Then E[ I i ] > 1 1 k for all i {1,..., k}. By linearity of expectations, E[ I I k ] > k 1. Therefore, there exists a partition X 1 X k of such that each X i contains a petal of S. As this gives k disoint petals, we conclude that S contains a sunflower of size k. In light of Proposition 5, if the bound l!((t /pl in the Approximate Sunflower Theorem can be replaced by f(p, t l for any function f(p, t (which does not depend on l, then the bound l!(k 1 l of the Sunflower Theorem can be replaced by f( 1 k, ln kl. This would prove the notorious Sunflower Conecture (see [ASU13, Juk11]. The hypothesis that such a function f(p, t exists was named the Approximate Sunflower Conecture by Lovett and Zhang [LZ18]. The rest of this note contains the proof of Theorem 3. The key tool from probabilistic combinatorics is Janson s Inequality (a.k.a. the Extended Janson s Inequality. 2

3 Theorem 6 (Janson s Inequality [Jan90]. Let S be any set system over a set and let X be a random subset of such that events {v X} are independent over v. Let µ : P[ A X ], : P[ A 1 A 2 X ]. A S Then P[ ( A S A X ] exp( µ 2 /. (A 1,A 2 S 2 : A 1 A 2 (In many statements of this inequality, the definition of includes the condition A 1 A 2 in the summation; in this case, one writes µ 2 /(µ + instead of µ 2 /. Rather than l!((t /pl, we shall prove a stronger version of Theorem 3 with the bound c l (t/p l for a certain sequence of polynomials c l (t. Definition 7. Let c 0 (t, c 1 (t,... be the sequence of polynomials defined by c 0 (t : 1, ( l c l (t : t c (t for l 1. 0 For l 1, we have the explicit expression c l (t t k k1 0 0 < 1 < < k l k i1 ( i i 1. Lemma 8. For all t > 0, we have l!t l c l (t l!(t l. Proof. For the lower bound, we have ( ( ( l l 1 1 c l (t t l l!t l. l 1 l 2 0 For the upper bound, we have the following proof by induction that c l (t l!(1/ ln( 1 t + 1l : ( l c l (t t c (t t 0 0 (! 1 ln( 1 t + 1 ( 1 l l! ln( 1 t + 1 t 0 ( 1 l! ln( 1 t + 1 l t( 1 + (ln( 1 t + 1l (l! k0 (ln( 1 t + 1k k! ( 1 l l! ln( 1 t + 1. Finally, we use the fact that 1/ ln( 1 t + 1 < t for all t > 0. In light of Lemma 8, Theorem 3 follows from the following theorem. Theorem 9. For every S ( l with S cl (t/p l, there exists B ( <l such that P [ ( A S B A X ] < e t. X p 3

4 Proof. Induction on l. In the base case, let S with S t/p. We have P [ ( v S v / X ] (1 X p S < e p S e t. p For the induction step, let l 2 and let S ( l with S cl (t/p l. We consider two cases. Case 1: There exists {1,..., l 1} and B ( l such that SB c l (t/p l. By the induction hypothesis, there exists C ( <l such that Since (S B C S B C, we are done. let P [ ( A (S B C A X ] < e t. X p Case 2: For all {1,..., l 1} and B ( l, we have SB < c l (t/p l. As in Theorem 6, µ : P [ A X ], : X p A S It suffices to show that µ 2 / > t. First, we have the lower bound We next upper bound : Therefore, µ + 1 µ + 1 p 2l p 2l (A 1,A 2 S 2 : A 1 A 2 µ p l S c l (t. B ( B ( P [ A 1 A 2 X ]. X p {(A 1, A 2 S 2 : A 1 A 2 B} < µ + p l c l (t 1 µ + p l µ + p l S µ 0 µ 2 1 B ( c l (t 1 A S c (t. S B ( l c l (t > µ l 1 0 c (t c l (t l 1 0 c (t t. Janson s Inequality now yields the desired bound P X p [ ( A S A X ] < e t. 4

5 Remark 10. This bound on is essentially tight. For i {1,..., l} and C ( i, instead of upper bounding the number of pairs (A 1, A 2 S 2 with A 1 A 2 C by S C 2 (in our bound on, we can instead use inclusion-exclusion to get an equality: {(A 1, A 2 S 2 : A 1 A 2 C} ( 1 i. i B ( : C B This gives the following exact expression for : i1 i1 p 2l i p 2l i 1 i1 1 C ( i C ( i i {(A 1, A 2 S 2 : A 1 A 2 C} ( 1 i ( p 2l i ( 1 i i p 2l B ( : C B B ( ( ( p i i i1 B ( p 2l ( (1 p ( p 1 B (. For small p, the value of (1 p ( p is very close to 1. Even in the case p 1 2, we get no significant improvement; in this case we have (1/2 2l. {1,3,5,...,2 l/2 1} B ( This allows us to replace c l (t in Theorem 3 with the polynomial d l (t k1 t k 0 0 < 1 < < k l : i i 1 is odd for all i {1,...,k} However, d l (t is still lower bounded by l!t l for t > 0. For this reason, it appear that any improvement to Theorem 3 beyond l!t l will require a substantially different proof technique. k i1 ( i i 1. 5

6 References [ASU13] [ER60] Noga Alon, Amir Shpilka, and Christopher Umans. On sunflowers and matrix multiplication. computational complexity, 22(2: , Paul Erdős and Richard Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, 1(1:85 90, [GMR13] Parikshit Gopalan, Raghu Meka, and Omer Reingold. DNF sparsification and a faster deterministic counting algorithm. Computational Complexity, 22(2: , [Jan90] Svante Janson. Poisson approximation for large deviations. Random Structures and Algorithms, 1(2: , [Juk11] [LLZ18] Stasys Jukna. Extremal combinatorics: with applications in computer science. Springer, Xin Li, Shachar Lovett, and Jiapeng Zhang. Sunflowers and quasi-sunflowers from randomness extractors. In APPROX-RANDOM, volume 116 of LIPIcs, pages 51:1 13, [LSZ18] Shachar Lovett, Noam Solomon, and Jiapeng Zhang. Unpublished work, [LZ18] Shachar Lovett and Jiapeng Zhang. DNF sparsification beyond sunflowers. ECCC preprint TR18-190, [Ros14] Benamin Rossman. The monotone complexity of k-clique on random graphs. SIAM Journal on Computing, 43(1: ,