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Depth through Breadth (or, why should we go to talks in other areas). Avi Wigderson IAS, Princeton. y Bob. x Alice. Are we still one community? Is there a connection between?. E-commerce / Algorithmic Game Theory Quantum Computing Circuit Complexity Optimization

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Depth through breadth or why should we go to talks in other areas

Depth through Breadth(or, why should we go to talks in other areas)

Avi Wigderson

IAS, Princeton


Are we still one community is there a connection between

y

Bob

x

Alice

Are we still one community?Is there a connection between?

  • E-commerce / Algorithmic Game Theory

  • Quantum Computing

  • Circuit Complexity

  • Optimization

  • VLSI & Distributed Computing

    Yes! e.gCommunication Complexity [Yao]


Combinatorial auctions
Combinatorial Auctions

Task: find partition

[k]= S1 S2… Sn

Max B1 (S1)+B2 (S2)+…+ Bn (Sn)

Seller: Goods {1,2,3,…,k}=[k]

BUYERSB1 B2 B3 …… Bn

BUNDLES

 0 0 0 0

{1} 2 5 0 7

{2} 1 0 4 4

{k} 1 13 3 9

{1,2} 4 12 4 8

{k-1,k} 11 24 3 16

[k] 15 72 66 34

Basic Question:

Can they find it efficiently

Polytime (k,n)

Thm[Nisan,Segal ’01]: No!

Time  exp(k)


Combinatorial auctions1
Combinatorial Auctions

Task: find partition

[k]= SA SB

Max A(SA)+B(SB)

Goods {1,2,3,…,k}=[k]

BUYERSAliceBob

BUNDLES

 0 0

{1} 2 0

{2} 1 4

{k} 1 3

{1,2} 4 4

{k-1,k} 11 3

[k] 15 66

Basic Question:

Can they find it efficiently

Polytime (k)

Thm[Nisan,Segal ‘01]: No!

Time  Communication  exp(k)


Combinatorial auctions2
Combinatorial Auctions

Task: find partition

[k]= S  Sc

Max A(S)+B(Sc)

Goods {1,2,3,…,k}=[k]

BUYERSAliceBob

BUNDLES BUNDLES

 0 1 [k]

{1} 0 1 [k]\{1}

{2} 1 1 [k]\{2}

{k} 1 0 [k]\{k}

{1,2} 1 1 [k]\{1,2}

{k-1,k} 0 0 [k]\{k-1,k}

[k] 1 0 

Thm[Nisan,Segal ‘01]: No!

Communication  exp(k)

Proof:

Max A(S)+B(Sc)=2 iff

1-bundles are disjoint!

Use disjointness lower bd:

Communication  exp(k)

(even probabilistic and nondeterministic!)


Quantum query complexity
(Quantum) Query Complexity

Compute f:{0,1}n{0,1} (with prob .99)

Resource: # of queries Q(f) to input bits

Pi(x) = Prob [ Alg accesses xi ]

Thm[Ambainis ‘01]: A: f(x)=0 B: f(y)=1

1/n 

A(x)=B(y)=i & xiyi

Prob[ ]  .98/Q(f)

f=OR [Grover search] x=0, y=ej for random j


Formula size

A: x=101

B: y=110

A: f(x)=0

B: f(y)=1

x1

x3

x1

x2

x2

x3

Formula Size

Compute f:{0,1}n{0,1}

Resources: size, depth

Thm[Karchmer-Wigderson ‘88]:

Pf: find i such that xiyi

Then cc(Pf) = depth (f)

  • Lower bounds on size of

  • Monotone formulae

  • Cutting Planes proofs

  • LOGSPACE  P via

  • information theory


Vlsi distributed computing

A

B

x1

x2

x3

f

VLSI & Distributed Computing

Compute f:{0,1}n{0,1}

Resources: Area, Time

Thm:[Aho,Ullman,

Yannakakis ‘83]

(Area)(Time)  cc’(f)

 (n)


Projecting linear programs
Projecting Linear Programs

Thm[Khachian ‘80]: Linear Programming  P

Fact: TSP is a linear program

Problem: Exponentially many facets (inequalities)

Idea: Write TSP polytope as a projection of another, with few facets

Claim[Swart ‘86]: P=NP via LP1 (with n8 vars)

Ref1: Bug in LP1

Claim[Swart ‘87]: P=NP via LP2 (with n10 vars)

Ref2: Bug in LP2

Thm[Yannakakis ‘88]: Swart’s approach must fail!


Projecting linear programs1
Projecting Linear Programs

Thm[Yannakakis]: Let LP be any program.

Set up the following CC problem hLP

A’s inputs: facets of LP

B’s inputs: vertices of LP

hLP(f,v)=1 iff v is not on f

hLP(f,v)=0 iff v is on f

If LP is the projection of LP’ then

#facets (LP’) exp( ncc(hLP) )

/ valid inequalities

/ feasible points


Multi party communication complexity

x

y

Number on

Forehead

Model

[Chandra, Furst, Lipton ‘83]

f(x,y,z)

z

Multi-party Communication Complexity

Branching Programs l.b.’s [Chandra, Furst, Lipton]

Turing machine l.b.’s [Babai, Nisan, Szegedy]

Threshold circuit l.b.’s [Goldman, Hastad]

ACC0 NC1 ? [Yao]

Space pseudorandom gen [Babai, Nisan, Szegedy]


The story of
The story of

Interactive

Proofs


NP: efficient

proofs

IP

[B,GMR]

Circuit

Complexity

Proof

Complexity

Dist Comp

Internet

#PIP [LFKN]

IP=PSPACE [S]

#PIP [LFKN]

IP=PSPACE [S]

#PIP [LFKN]

IP=PSPACE [S]

PCP(log n,1)=NP

[AS,ALMSS]

Optimization

Approx

Randomized

Computation

Interactive

Proofs

Program

Checking

Property

Testing

Coding

Theory

Cryptography

Zero-Knowledge

PCP

[BFLS,FGLSS]

MIP

[BGKW]

MIP

[BGKW]

MIP

[BGKW]

MIP=NEXP

[BFL]

Permanent

MIP [N]

Per is RSR

[L,BF]

Streaming,

Sublinear

Algorithms

Permanent

#P-complete [V]

PH-hard [T]

Approx [JSV]


What is the glue
What is the glue?

Models, like

Communication Complexity

  • E-commerce

  • Quantum Computing

  • Circuit Complexity

  • Distributed Computing

  • Optimization

Techniques, like

Pairwise Independence

  • Data Structures

  • Derandomization

  • Learning Theory

  • Cryptography

  • BPPPH, AM=IP, UPP

Problems, like

Permanent

  • Structural Complexity

  • Statistical Physics

  • Comb Optimization

  • Arithmetic Circuits

  • Interactive Proofs

Algorithms, like

Iterative alg for LPs

  • Boosting of learning algs

  • Hard-core sets

  • On-line routing

  • Congestion control TCP/IP

  • Parallel matching alg


What is the glue1
What is the glue?

People, like

Les Valiant

  • Circuit Complexity

  • Parallel Computation

  • Learning

  • Neural Computation

  • Quantum algorithms

Objects, like

Expanders

  • Data Structures

  • Derandomization

  • Networks

  • Coding Theory

  • Mathematics

Language, or Level at which we conceptualize

  • Asymptotic analysis

  • Adversaries(worst-case & amortized analysis)

  • Generality

  • Connections/Reductions

Subject: Computation

  • Biological processes(DNA, cell, brain, populations…)

  • Physical processes (atoms, weather, galaxies)

  • Internet, Stock Market

  • Proofs


Stoc focs culture
STOC/FOCS culture

  • Frequent, well attended

  • Open, inclusive (even imperialistic)

  • Tolerant to new (weird?) ideas

  • Student friendly, interactive

  • Dynamic, (too?) fast changing

  • Driving (deadline generated papers)

  • Heterogeneous, many diverse topics

  • No parallel sessions (I wish), so we can go to talks in other areas


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