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

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

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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

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 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 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

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 & xiyi

Prob[ ] .98/Q(f)

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

A: x=101

B: y=110

A: f(x)=0

B: f(y)=1

x1

x3

x1

x2

x2

x3

Formula SizeCompute f:{0,1}n{0,1}

Resources: size, depth

Thm[Karchmer-Wigderson ‘88]:

Pf: find i such that xiyi

Then cc(Pf) = depth (f)

- Lower bounds on size of
- Monotone formulae
- Cutting Planes proofs
- LOGSPACE P via
- information theory

B

x1

x2

x3

f

VLSI & Distributed ComputingCompute f:{0,1}n{0,1}

Resources: Area, Time

Thm:[Aho,Ullman,

Yannakakis ‘83]

(Area)(Time) cc’(f)

(n)

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 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

y

Number on

Forehead

Model

[Chandra, Furst, Lipton ‘83]

f(x,y,z)

z

Multi-party Communication ComplexityBranching 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]

proofs

IP

[B,GMR]

Circuit

Complexity

Proof

Complexity

Dist Comp

Internet

#PIP [LFKN]

IP=PSPACE [S]

#PIP [LFKN]

IP=PSPACE [S]

#PIP [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?

Models, like

Communication Complexity

- E-commerce
- Quantum Computing
- Circuit Complexity
- Distributed Computing
- Optimization

Techniques, like

Pairwise Independence

- Data Structures
- Derandomization
- Learning Theory
- Cryptography
- BPPPH, AM=IP, UPP

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 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

- 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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