Analytical Combinatorics

1. Analytical Combinatorics

2. Boolean Functions • Def: ABoolean function Power set of [n] Choose the location of -1 Choose a sequence of -1 and 1

3. Noise Sensitivity • The values of every variables may, independently, change with probability  • It turns out: no Boolean f is robust under noise --that is, would, on average, change w.p. <sqrt()-- unless the outcome is almost always determined by very few variables (disregarding all but exp(1/ ))

4. Voting and influence • Def: theinfluence ofi on f is the probability, over a random input x, that f changes its value when i is flipped -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 1 -1

5. -1 1 -1 -1 ? 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 • Majority:{1,-1}n {1,-1} • Theinfluence of i on Majority is the probability, over a random input x, Majority changes with i • this happens when half of the n-1 coordinate (people) vote -1 and half vote 1. • i.e.

6. -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 • XOR: {1,-1}n {1,-1} Always changes the value of parity

7. -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 • Dictatorshipi:{1,-1}20 {1,-1} • Dictatorshipi(x)=xi • influence of i on Dictatorshipi= 1. • influence of ji on Dictatorshipi=0.

8. Total-Influence (Average Sensitivity) • Def: theTotal-Influence off(as) is the sum of influences of all variables i[n]: • as(Majority) = O(n½) • as(Parity) = n • as(dictatorship) =1

9. Representing f as a Polynomial • What would be the monomials over x  P[n] ? • All powers except 0 and 1 cancel out! • Hence, one for each characterS[n] • These are all the multiplicative functions

10. Fourier-Walsh Transform • Consider all characters • Given any functionlet the Fourier-Walsh coefficients of f be • thus f can be described as

11. Norms Def: (Expectation) norm on the function Thm [Parseval]: for a Boolean f

12. SimpleObservations • Def: • Claim:For any function f whose range is {-1,0,1}:

13. Variables` Influence • Recall: influence of an index i [n] on a Boolean function f:{1,-1}n {1,-1} is • Which can be expressed in terms of the Fourier coefficients of fClaim: • And the as:

14. Expectation and Variance • Claim: • Hence, for any f

15. Heuristics: Hardness Amplification • Claim: • Hence, for any f

16. Monotone Substitute for XOR • Claim:for monotone functions I[f] < sqrt n • Find a monotone function f so that almost all input settings x have sqrt n pivotal bits

17. Percolation Each edge occurs w/probability½

18. Graph properties Def: A graph property is a subset of graphs invariant under isomorphism. Def: a monotone graph property is a graph property P s.t. • If P(G) then for every super-graph H of G (namely, a graph on the same set of vertices, which contains all edges of G) P(H) as well. P is in fact a Boolean function:P: {-1, 1}V2{-1, 1}

19. Examples of graph properties • G is connected • G is Hamiltonian • G contains a clique of size t • G is not planar • The clique number of G is larger than that of its complement • The diameter of G is at most s • ... etc . • What is the influence of different e on P?

20. Erdös–Rényi G(n,p)Graph TheErdös-Rényidistribution of random graphs Put an edge between any two vertices w.p.p

21. Definitions • P – a graph property • p(P) - the probability that a random graph on n vertices with edge probability p satisfies P. • GG(n,p) - G is a random graph of n vertices and edge probability p.

22. Def: Sharp threshold • Sharp threshold in monotone graph property: • The transition from a property being very unlikely to it being very likely is very swift. G satisfies property P GDoes not satisfies property P

23. Thm: every monotone graph property has a Sharp Threshold[FK] • Let P be any monotone property of graphs on n vertices . If p(P) >  then q(P) > 1- for q=p + c1log(½)/logn Proof idea: show asp’(P), for p’>p, is high

24. weight characters …-5 -3 -1 1 3 5… Concentrated • Def: the restrictionof f to  is • Def: f is a concentrated function if >0,  of poly(n/) size s.t. • Thm [Goldreich-Levin, Kushilevitz-Mansour]: f:{0,1}n{0,1} concentrated is learnable • Thm [Akavia, Goldwasser, S.]: over any Abelian group f:GnG

25. -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 Juntas • A function is a J-junta if its value depends on only J variables. 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 • A Dictatorship is 1-junta -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1

26. -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 Juntas • A function is a J-junta if its value depends on only J variables. 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 • Thm[Fischer, Kindler, Ron, Samo., S]: Juntas are testable • Thm[Kushilevitz, Mansour; Mossel, Odonel]: Juntas are learnable

27. I  - Noise sensitivity Choose a subset, I, of variables Each var is in the set with probability  Redraw each value of the subset, I with probability p • The noise sensitivity of a function f is the probability that f changes its value when redrawing a subset of its variables according to the p distribution. What is the new value of f? -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1

28. Choose a subset (I) of variables Each var is in the set with probability  Junta I Noise sensitivity and juntas Redraw each value of the subset (I) with probability p • Juntas are noise insensitive (stable) Thm[Bourgain; Kindler & S]: Stable B.f. are JuntasThm[MOO]: Majority Stablest if low Inluencei What is the new value of f? W.H.P STAY THE SAME -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1

29. Freidgut Theorem Thm: any Boolean f is an [, j]-junta for Proof: • Specify the junta J • Show the complement ofJ has little influence

30. Coding Theory • Def: a binary code is C  {-1, 1}t • Rate: log|C|/t • Distance: D such that for any x, yCH(x, y) ≥ D • A string of length 2nis a Boolean function {-1, 1}n {-1, 1}, hence a code is a class of Boolean functions • Hadamard code: all characters • Long Code: all dictatorships

31. Testing Codes (PCP related) Def (a code list-test): given an f, probe it in a constant number of entries, and • accept (almost) always if f is legal • reject w.h.p if fdoes not have a positive correlation with any legal code-word • If not rejected, there is a short list of legal code-words with positive correlation

32. Hadamard Test Given a Boolean f, choose random x and y; check that f(x)f(y)=f(xy) Prop(completeness): a legal Hadamard word (a character) always passes this test

33. Long-Code Test Given a Boolean f, choose random x and y, and choose z; check thatf(x)f(y)=f(xyz) Prop(completeness): a legal long-code word (a dictatorship) passes this test w.p. 1-

34. Testing Long-code Def(a long-code list-test): given a code-word f, probe it in a constant number of entries, and • accept almost always if f is a monotone dictatorship • reject w.h.p if fdoes not have a sizeable fraction of its Fourier weight concentrated on a small set of variables, that is, if a semi-JuntaJ[n] s.t. Note: a long-code list-test, distinguishes between the case f is a dictatorship, to the case f is far from a junta.

35. Motivation – Testing Long-code • The long-code list-test are essential tools in proving hardness results. • Hence finding simple sufficient-conditions for a function to be a junta is important.

36. Open Questions • Entropy Conjecture [FK] • Classify functions that are closed under a large subgroup of Sn • Hardness of Approximation: • Coloring a 3-colorable graph with fewest colors • Graph Properties: find real sharp-thresholds for properties • Circuit Complexity: switching lemmas • Mechanism Design: show a non truth-revealing protocol in which the pay is smaller (Nash equilibrium when all agents tell the truth?) • Learning: by random queries • Apply Concentration of Measure techniques to other problems in Complexity Theory