Download presentation

Presentation is loading. Please wait.

Published byAutumn Quinn Modified over 3 years ago

1
Parikshit Gopalan Georgia Institute of Technology Atlanta, Georgia, USA.

2
Explicit Ramsey Graph Constructions [Erdös] : There exists a graph G on 2 n vertices with (G), (G) · 2n. Probabilistic Method. $100 for explicit construction. [Ramsey] : Every graph on 2 n vertices has either an independent set or a clique of size n/2. Easy to construct G on 2 n vertices with (G), (G) · 2 n/2.

3
Alternate View Constructing Ramsey graphs: Two color the edges of K n so that there are no large monochromatic cliques. Constructing Multicolor Ramsey graphs: Color the edges of K n using t colors so that there are no large monochromatic cliques.

4
A Brief History of Explicit Constructions [Frankl-Wilson] : Gives (G), (G) · 2 n. Extremal set theory. [Grolmusz] : Same bound, multicolor graphs. Polynomial representations of the OR function. [Alon] : Similar to Frankl-Wilson, multicolor graphs. Polynomial representations of graphs. [Barak-Rao-Shaltiel-Wigderson] : (G), (G) · 2 n. Extractors and pseudorandomness.

5
Lower Bounds for AC 0 [m] Problem: Find a function that cannot be computed by polynomial size, constant depth circuits with AND, OR and Mod-m gates. For Mod-p gates (p is prime) [Razborov, Smolensky] With p = 2, the Mod-3 function is hard. For Mod-m gates (m is composite, say 6) With m = 6, is the Mod-5 function hard ? No lower bounds known for any natural function.

6
Polynomial Representations of Boolean functions Def: P(X 1,…,X n ) over Z m represents f: {0,1} n ! {0,1} if f(x) f(y) ) P(x) P(y) mod m Lower bounds for AC 0 [m]. Prime Case: [Razborov, Smolensky] : Small circuits Low-degree polynomials. Prove degree lower bounds. Composite Case: Low-degree polynomials ) Small circuits Degree lower bounds over Z m. (Simpler problem?)

7
Representing the OR function Problem: What is the degree of OR mod m ? For p prime: (n). For m composite (say 6): Conjecture: (n) [Barrington] O(n 1/2 ) upper bound. [Barrington-Beigel-Rudich] (log n) lower bound. [Barrington-Tardos] Can asymmetry help compute a symmetric function? [Barrington-Beigel-Rudich, Grolmusz, Tsai, Barrington- Tardos, Green, Alon-Beigel, Bhatnagar-G.-Lipton, Hansen]

8
A Connection [Grolmusz] Problem: Let F be a family of subsets S i of [n] where |S i | = 0 mod m |S i Å S j | 0 mod m How large can F be? Grolumsz: If m = 6, |F| can be superpolynomial in n. Uses O(n) degree OR polynomial of BBR. Gives a Ramsey graph matching FW. Better OR polynomials ) Better graphs.

9
Our Results New view of OR representations. Simple Ramsey construction from OR representations. Unifies Frankl-Wilson, Grolmusz, Alon. All based on O(n) symmetric OR polynomials. Consequences : Insight from complexity: Asymmetry versus Symmetry Extends to multicolor Ramsey graphs. Improved bounds for restricted set systems.

10
Outline of This Talk I Ramsey graphs from OR representations. New view of OR representations. Sample constructions. Ramsey graphs. II Limitations to Symmetric Constructions.

11
Outline of This Talk I Ramsey graphs from OR representations. New view of OR representations. Sample constructions. Ramsey graphs. II Limitations to Symmetric Constructions.

12
OR Representations New view of an OR representation: Two polynomials s.t. the union of their zero sets is {0,1} n \ {0}.

13
P = 0Q = 0

14
OR Representations New view of an OR representation: Two polynomials. Union of their zero sets is {0,1} n \ {0}. Degree of representation = max(deg(P), deg(Q)). Both polynomials mod p. P mod p, Q mod q. Prime Representations Both polynomials mod p a. Prime-power representations (n) O(n) [BBR, Alon] O(n) [FW] All give O(n) degree symmetric polynomials.

15
Alons Construction Choose p ¼ q, and let n = pq -1. Let P(X 1,…,X n ) = 1 – ( X i ) p-1 mod p Indicator for X i being divisible by p. Let Q(X 1,…,X n ) = 1 – ( X i ) q-1 mod q Indicator for X i being divisible by q. Both are 1 only for (0,…,0). Degree of the construction is max(p,q) = O(n). [BBR94] Take p = 2, q = 3. Special cases of OR representations modulo pq. [Frankl-Wilson] Take n = p 2 -1. Both polynomials modulo powers of p.

16
The Frankl-Wilson Construction Let n = p 2 -1. P(X 1,…,X n ) stays the same. Indicator for X i being divisible by p. Remaining weights are 0, p, 2p, …, p 2 -p. Q(X 1, …, X n ) = ( X i –p)...( X i – p 2 + p)mod p p Both are 1 only for (0,…,0). Degree of the construction is p = O(n). Does not use Chinese Remaindering. Extends to multicolor graph constructions.

17
Plug in X = x 1 : P(0,…,0) 0, P(x 1 © x 2 ) = 0, …, P(x 1 © x k ) = 0. The Ramsey Graph Construction Ramsey Construction: Vertices: {0,1} n. Edges: Add edge (x,y) if P(x © y) = 0. Thm: Degree d OR representation gives (G), (G) · n d. Consider a clique: x 1, …, x k We have: P(x i © x j ) = 0. Claim : Polynomials P(X © x 1 ), …, P(X © x k ) are LI. Dimension of vector space O(n d ). Hence k · n d. Bound size of (G) using Q(X 1,…,X n ).

18
The Ramsey Graph Construction Ramsey Construction: Vertices: {0,1} n. Edges: Add edge (x,y) if P(x © y) = 0. Thm: Degree d OR representation gives (G), (G) · n d. Proof by the linear algebra method [Babai-Frankl]. Plugging in d = O(n) gives a bound of 2 n. Lower degree ) better graphs.

19
Outline of This Talk I Ramsey graphs from OR representations. New view of OR representations. Sample constructions. Ramsey graphs construction. II Limitations to Symmetric Constructions.

20
Limitations to Symmetric Constructions Thm : (n) lower bound for symmetric polynomials. For any OR representation, deg(P) £ deg(Q) = (n). Symmetry vs asymmetry question applies to Ramsey graph constructions.

21
P mod p, Q mod q. [BBR, Alon] Gives a representation of OR over Z pq. Known lower bound: (n/pq). When n < pq [Alon] … X i represents OR mod pq. Both polynomials mod p a.[FW] Based on interpolation algorithm mod p a [G06]. Thm : (n) lower bound for symmetric polynomials. Limitations to Symmetric Constructions

22
High-Level Idea 1.Algebraic Step: Characterize zero-sets of low- degree multivariate polynomials. 2.Combinatorial Step: Show that there is no good partition of {0,1} n. Symmetry : Replace multivariate by univariate. Replace {0,1} n by {0,…,n}

23
Partition Problem Adversary gets number n. Picks 1. Primes p and q where p ¢ q > n. 2. A µ {1,…, p-1} and B µ {1, …, q-1} Every x 2 {1, …, n} is covered by A or B. Minimize |A| ¢ |B|. x mod p lies in A

24
Partition Problem Adversary gets number n. Picks 1. Primes p and q where p ¢ q > n. 2. A µ {1,…, p-1} and B µ {1, …, q-1} Every x 2 {1, …, n} is covered by A or B. Minimize |A| ¢ |B|. 1 2 3 4 1 2 3 4 5 6 p = 5, q = 7, n = 12 112…

25
Partition Lemma Trivial Solutions : A = {1,…, p-1} and B = {p, 2p, …, } A = {q, 2q, …} and B = {1, …, q-1} Gives |A| ¢ |B| = n. Better solutions ) Better OR representations. Partition Lemma: In any solution, |A| ¢ |B| ¸ n/8.

26
Symmetry vs. Asymmetry Do low degree OR polynomials exist? Conjecture [Barrington-Beigel-Rudich]: No! (for representations mod 6) Symmetric polynomials for Symmetric functions. CRT. Hard explicit construction problem ? Symmetric polynomials give graphs on {0,1} n based on distances. Q : Are such graphs not good Ramsey graphs?

28
Prime-Power Representations Algebraic Step : Given a univariate polynomial P(X) mod p a which is 0 at ( 1, …, k ). Give a lower bound on deg(P). Low degree polynomials mod p a have many roots. Eg: X 3 = 0 mod 27 (0,3,6 …, 24) 9 X = 0 mod 27(0,3,6 …, 24) Structure theorem for zero sets of low-degree polynomials. (Builds on interpolation algorithm [G06])

29
603 mod 9 01 2 mod 3 09121114 Roots of low degree polynomials are clustered together. Let m = 27. I Zero-Sets of low degree polynomials

Similar presentations

OK

Extracting Randomness From Few Independent Sources Boaz Barak, IAS Russell Impagliazzo, UCSD Avi Wigderson, IAS.

Extracting Randomness From Few Independent Sources Boaz Barak, IAS Russell Impagliazzo, UCSD Avi Wigderson, IAS.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on indian textile industries in mauritius Download ppt on civil disobedience movement in 1930 Ppt on applied operational research pdf Ppt on personal financial planning Ppt on use of computer in animation Ppt on endangered species of animals Ppt on project proposal writing Ppt on intelligent manufacturing in industrial automation Ppt on c language arrays Ppt on school life and college life