Analysis of Boolean Functions and Complexity Theory Economics Combinatorics …

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

Influential People The theory of the Influence of Variables on Boolean Functions [KKL,BL,R,M], has been introduced to tackle Social Choice problems and distributed computing. It has motivated a magnificent body of work, related to Sharp Threshold [F, FG] Percolation [BKS] Economics: Arrow’s Theorem [K] Hardness of Approximation [DS] Utilizing Harmonic Analysis of Boolean functions… And the real important question:

Where to go for Dinner? The alternatives Diners would cast their vote in an (electronic) envelope The system would decide – not necessarily by majority… And what if someone (in Florida?) can flip some votes Power influence

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

Def: the influence of i on f is the probability, over a random input x, that f changes its value when i is flipped Def: the influence of i on f is the probability, over a random input x, that f changes its value when i is flipped influence

Functions as Vector-Spaces f f * * -1* 1* 11* 11-1* -1-1* -11* -11-1* -111* * -1-11* 111* 1-1* 1-1-1* 1-11* f f 2n2n 2n2n * * -1* 1* 11* 11-1* -1-1* -11* -11-1* -111* * -1-11* 111* 1-1* 1-1-1* 1-11*

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

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

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

influence of i on Dictatorship i = 1. influence of i on Dictatorship i = 1. influence of j  i on Dictatorship i = 0. influence of j  i on Dictatorship i = 0. Dictatorship i :{1,-1} 20  {1,-1} Dictatorship i :{1,-1} 20  {1,-1} Dictatorship i (X)=x i Dictatorship i (X)=x i

Variables` Influence The influence of a coordinate i  [n] on a Boolean function f:{1,-1} n  {1,-1} is 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.

Variables` Influence Average­ Sensitivity of f (AS) - The sum of influences of all coordinates i  [n]. Average­ Sensitivity of f (AS) - The sum of influences of all coordinates i  [n]. Average­ Sensitivity of f is the expected number of coordinates, for a random input x, flipping of which changes the value of f. Average­ Sensitivity of f is the expected number of coordinates, for a random input x, flipping of which changes the value of f.

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

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

Fourier-Walsh Transform Consider all characters Consider all characters Given any function let the Fourier-Walsh coefficients of f be Given any function let the Fourier-Walsh coefficients of f be thus f can be described as thus f can be described as

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

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

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

Inner Product Recall Recall Inner product (normalized) Inner product (normalized)

Simple Observations Claim: Claim: For any function f whose range is {-1,0,1}: For any function f whose range is {-1,0,1}:

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

Average Sensitivity Def: the sensitivity of x w.r.t. f is Def: the sensitivity of 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)). 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. Edge whose vertices are colored with the same color is called monotone. The average sensitivity is the number of edges whom are not monotone.. The average sensitivity is the number of edges whom are not monotone..

average sensitivity of Majority is the expected number of coordinates, for a random input x, flipping of which changes the value of Majority. average sensitivity of Majority is the expected number of coordinates, for a random input x, flipping of which changes the value of Majority. Majority :{1,-1} 19  {1,-1} Majority :{1,-1} 19  {1,-1} 1?

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

influence of i on Dictatorship i = 1. influence of i on Dictatorship i = 1. influence of j  i on Dictatorship i = 0. influence of j  i on Dictatorship i = 0. Dictatorship i :{1,-1} 20  {1,-1} Dictatorship i :{1,-1} 20  {1,-1} Dictatorship i (X)=x i Dictatorship i (X)=x i

Average Sensitivity Claim: Claim: Proof: Proof:

When AS(f)=1 Def: f is a balanced function if Def: f is a balanced function if THM: f is balanced and as(f)=1  f is dictatorship. THM: f is balanced and as(f)=1  f is dictatorship. Proof: Proof:   x, sens(x)=1, and as(f)=1 follows.   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. 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

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

First Passage Percolation

Choose each edge with probability ½ to be a and ½ to be b

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

First Passage Percolation Choose each edge with probability ½ to be 1 and ½ to be 2

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

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

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.

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

Graph property Every Monotone Graph Property has a sharp threshold

A graph property is a property of graphs which is closed under isomorphism. A graph property is a property of graphs which is closed under isomorphism. monotone graph property : monotone graph property : Let P be a graph property. Let P be a graph property. Every graph H on the same set of vertices, which contains G as a sub graph satisfies P as well. Every graph H on the same set of vertices, which contains G as a sub graph satisfies P as well. Graph property

Examples of graph properties G is connected G is connected G is Hamiltonian G is Hamiltonian G contains a clique of size t G contains a clique of size t G is not planar G is not planar The clique number of G is larger than that of its complement The clique number of G is larger than that of its complement the diameter of G is at most s the diameter of G is at most s... etc.... etc.

Erdös – Rényi Graph Model Erdös - Rényi for random graph Model Erdös - Rényi for random graph Choose every edge with probability p Choose every edge with probability p

Erdös – Rényi Graph Model Erdös - Rényi for random graph Model Erdös - Rényi for random graph Choose every edge with probability p Choose every edge with probability p

Every Monotone Graph Property has a sharp threshold Ehud Friedgut & Gil Kalai

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

Main Theorem Let GNP be any monotone property of graphs on n vertices. Let GNP be any monotone property of graphs on n vertices. If  p (GNP) >  then  q (GNP) > 1-  for q = p + c 1 log(1/2  )/logn absolute constant

Example-Max Clique Consider G  G(n,p). Consider G  G(n,p). The length of the interval of probabilities p for which the clique number of G is almost surely k (where k  log n) is of order log -1 n. The length of the interval of probabilities p for which the clique number of G is almost surely k (where k  log n) is of order log -1 n. The threshold interval: The transition between clique numbers k-1 and k. The threshold interval: The transition between clique numbers k-1 and k. Probability for choosing an edge Number of vertices

The probability of having a (k + 1)-clique is still small (  log -1 n). The probability of having a (k + 1)-clique is still small (  log -1 n). The value of p must increase byclog -1 n before the probability for having a (k + 1)- clique reaches  and another transition interval begins. The value of p must increase by clog -1 n before the probability for having a (k + 1)- clique reaches  and another transition interval begins. The probability of having a clique of size k is 1-  The probability of having a clique of size k is 

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

Conjecture Let GNP be any monotone property of graphs on n vertices. If  p (GNP) >  then  q (GNP) > 1-  for q = p + clog(1/2  )/log 2 n Let GNP be any monotone property of graphs on n vertices. If  p (GNP) >  then  q (GNP) > 1-  for q = p + clog(1/2  )/log 2 n

Graph property Every Monotone Graph Property has a sharp threshold

A graph property is a property of graphs which is closed under isomorphism. A graph property is a property of graphs which is closed under isomorphism. hereditary : hereditary : Let P be a monotone graph property; that is, if a graph G satisfies P Let P be a monotone graph property; that is, if a graph G satisfies P Every graph H on the same set of vertices, which contains G as a sub graph satisfies P as well. Every graph H on the same set of vertices, which contains G as a sub graph satisfies P as well. Graph property

Hereditary in 3-colorable graphs

Examples of graph properties G is connected G is connected G is Hamiltonian G is Hamiltonian G contains a clique of size t G contains a clique of size t G is not planar G is not planar The clique number of G is larger than that of its complement The clique number of G is larger than that of its complement the diameter of G is at most s the diameter of G is at most s G admits a transitive orientation G admits a transitive orientation... etc.... etc.

Erdös – Rényi Graph Model Erdös - Rényi for random graph Model Erdös - Rényi for random graph Choose every edge with probability p Choose every edge with probability p

Erdös – Rényi Graph Model Erdös - Rényi for random graph Model Erdös - Rényi for random graph Choose every edge with probability p Choose every edge with probability p

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

Example – max clique Let G  G(n,p) Let G  G(n,p)

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

Mechanism Design Shortest Path Problem

Mechanism Design Problem N agents,bidders, each agent i has private input t i  T. Everything else in this scenario is public knowledge. N agents,bidders, each agent i has private input t i  T. Everything else in this scenario is public knowledge. The output specification maps to each type vector t= t 1 …t n a set of allowed outputs o  O. The output specification maps to each type vector t= t 1 …t n a set of allowed outputs o  O. Each agent i has a valuation for his items: V i (t i,o) = outcome for the agents. Each agent wishes to optimize his own utility. Each agent i has a valuation for his items: V i (t i,o) = outcome for the agents. Each agent wishes to optimize his own utility. Objective: minimize the objective function, the total payment. Objective: minimize the objective function, the total payment. Means: protocol between agents and auctioneer. Means: protocol between agents and auctioneer.

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

Vickery-Groves-Clarke (VGC)

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

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

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

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

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

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

Freidgut Theorem Thm: any Boolean f is an [ , j]-junta for Proof: 1. Specify the junta J 2. Show the complement of J has little influence

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

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

Altogether Lemma: Proof:

Altogether

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

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

Beckner/Nelson/Bonami Corollary Corollary 1: for any real f and 2≥r≥1 Corollary 2: for real f and r>2

Freidgut Theorem Thm: any Boolean f is an [ , j]-junta for Proof: 1. Specify the junta J 2. Show the complement of J has little influence

Altogether Beckner