Analysis of Boolean Functions and Complexity Theory Economics Combinatorics …

1. Analysis of Boolean FunctionsandComplexity TheoryEconomicsCombinatorics…

2. Introduction • Objectives: • To introduce Analysis of Boolean Functions and some of its applications. • Overview: • Basic definitions. • First passage percolation • Mechanism design • Graph property • … And more…

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

4. f 111* 11* 11-1* 1* 1-11* 1-1* 1-1-1* * -111* -11* -11-1* -1* -1-11* -1-1* -1-1-1* Functions as Vector-Spaces f 111* 11* 11-1* 1* 1-11* 1-1* 1-1-1* 2n * -111* -11* -11-1* -1* -1-11* -1-1* -1-1-1*

5. Functions’ Vector-Space • A functions f is a vector • Addition:‘f+g’(x) = f(x) + g(x) • Multiplication by scalar‘cf’(x) = cf(x)

6. Variable influence • f:{1,-1}20 {1,-1} -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 • influence of i on f is the probability that f changes its value when i is flipped in a random input x.

7. -1 1 -1 -1 ? 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 • Majority :{1,-1}19 {1,-1} • influence of i on Majority is the probability that Majority changes its value when i is flipped in a random input x this happens when half of the n-1 coordinate (people) vote -1 and half vote 1. • i.e.

8. -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 • Parity :{1,-1}20 {1,-1} Always changes the value of parity

9. -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.

10. Variables` Influence • The influence of a coordinate i [n] on a Boolean function f:{1,-1}n {1,-1} is The influence of i on f is the probability, over a random input x, that f changes its value when i is flipped.

11. Variables` Influence • Average­ Sensitivityof f(AS) - The sum of influences of all coordinates i  [n]. • Average­ Sensitivityof fis theexpectednumber of coordinates, for a random input x, flipping of which changes the value of f.

12. Influence Influence Influence 2 1 3 example • majority for • What is Average Sensitivity ? • AS= ½+ ½+ ½= 1.5

13. Monomials • What would be the monomials over x  P[n]? • All powers except 0 and 1 disappear! • Hence, one for each characterS[n] • These are all the multiplicative functions

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

15. Norms • Def:Expectation norm on the function • Def:Summation Norm on its Fourier transform

16. Fourier Transform: Norm Norm: (Sum) Thm [Parseval]: Hence, for a Boolean f

17. We may think of the Transform as defining a distribution over the characters.

19. InnerProduct • Recall • Inner product(normalized)

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

21. 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:

22. Average Sensitivity • Def: thesensitivityof x w.r.t. f is • Thinking of the discrete n-dimensional cube, color each vertex n in color 1 or color -1 (color f(n)). • Edge whose vertices are colored with the same color is called monotone. • Theaverage sensitivityis the number of edges whom are not monotone..

23. Average Sensitivity • Claim: • Proof:

24. When AS(f)=1 • Def: f is abalancedfunction if • THM: f isbalanced andas(f)=1 f isdictatorship. • Proof: •  x, sens(x)=1, and as(f)=1 follows. • F is balanced since the dictator is 1 on half of the x and -1 on half of the x. because only x can change the value of f

25. If s s.t |s|>1 and then as(f)>1 When AS(f)=1 f is balanced •  • So f is linear • For i whose Only i has changed

26. First Passage Percolation • Consider the Grid • For each edge e of chooseindependentlywe = a or we = b, each with probability ½ 0< a < b < . • This induces a random metric on the vertices of • Proposition : The variance of the shortest path from the origin to vertex v is bounded by O( |v| log |v|). [BKS]

27. First Passage Percolation Choose each edge with probability ½ to be 1

29. Proof sketch

30. First Passage Percolation • Consider the Grid • For each edge e of chooseindependentlywe = 1 or we = 2, each with probability ½. • This induces a random metric on the vertices of • Proposition : The variance of the shortest path from the origin to vertex v is bounded by O( |v| log |v|).

31. Proof outline • Let G denote the grid • SPG– the shortest path in G from the origin to v. • Let denote the Grid which differ from G only on we i.e. flip coordinate e in G. • Set

32. Observation If e participates in a shortest path then flipping its value will increase or decrease the SP in 1 ,if e is not in SP - the SP will not change.

33. Proof cont. • And by [KKL] there is at least one variable whose influence was as big as (n/logn)

34. Mechanism Design Shortest Path Problem

35. Mechanism Design Problem • N agents,bidders, each agent i hasprivateinput tiT. Everything else in this scenario ispublicknowledge. • Theoutput specificationmaps to each type vector t= t1 …tn a set of allowed outputs oO. • Each agent i has avaluationfor his items: Vi(ti,o) = outcome for the agents.Each agent wishes to optimize his own utility. • Objective: minimize the objective function, the total payment. • Means: protocol between agents and auctioneer.

36. Truth implementation • The action of an agent consists of reporting its type, its true type. • Playing the truth is the dominating strategy • THM: If there exists a mechanism then there exists also a Truthful Implementation. • Proof: simulate the hypothetical implementationbased on the actions derived from the reported types.

37. Vickery-Groves-Clarke (VGC)

38. 10$ 10$ 50$ 50$ Mechanism Design for SP Always in the shortest path

39. Shortest Path using VGC • Problem definition: • Communication networkmodeled by a directed graph G and two vertices source s and target t. • Agents = edges in G • Each agent has a cost for sending a single message on his edge denote by te. • Objective: find the shortest (cheapest) path from s to t. • Means: protocol between agents and auctioneer.

40. Shortest Path using VGC • C(G)= costs along the shortest path (s,t) in G. • compute a shortest path in the G , at cost C(G) . • Each agent that participates in the SP obtains the payment she demanded plus [ C(G\e) – te ]. SP on G\e

41. 10$ 10$ 50$ 50$ How much will we pay?

42. -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 junta • 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

43. High vs. Low Frequencies Def: The section of a function f above k is and the low-frequency portion is

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

45. Specify the Junta Set k=(as(f)/), and =2-(k) Let We’ll prove: and let hence, J is a [,j]-junta, and |J|=2O(k)

46. High Frequencies Contribute Little Prop: k >> r log rimplies Proof: a character S of size larger than k spreads w.h.p. over all parts Ih, hence contributes to the influence of all parts.If such characters were heavy (>/4), then surely there would be more than j parts Ih that fail the t independence-tests

47. Altogether Lemma: Proof:

48. Altogether

49. Beckner/Nelson/Bonami Inequality Def: let Tbe the following operator on any f, Prop: Proof:

50. Beckner/Nelson/Bonami Inequality Def: let Tbe the following operator on any f, Thm: for any p≥rand≤((r-1)/(p-1))½