Depth through Breadth(or, why should we go to

talks in other areas)

Avi WigdersonIAS, Princeton

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

• E-commerce / Algorithmic Game Theory• Quantum Computing• Circuit Complexity• Optimization• VLSI & Distributed ComputingYes! e.g Communication Complexity [Yao]

x

Alice

y

Bob

Combinatorial Auctions Seller: Goods {1,2,3,…,k}=[k]

BUYERS B1 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

Task: find partition [k]= S1 S2… Sn

Max B1 (S1 ) +B2 (S2 ) +…+ Bn

(Sn ) Basic Question:Can they find it efficientlyPolytime (k,n)

Thm[Nisan,Segal ’01]: No!Time exp(k)

Combinatorial Auctions Goods {1,2,3,…,k}=[k]

BUYERS Alice Bob

BUNDLES 0 0 {1} 2 0 {2} 1 4 …{k} 1 3 {1,2} 4 4 …{k-1,k} 11 3 …[k] 15 66

Task: find partition [k]= SA SB

Max A(SA) +B (SB )

Basic Question:Can they find it efficientlyPolytime (k)

Thm[Nisan,Segal ‘01]: No!Time Communication exp(k)

Combinatorial Auctions Goods {1,2,3,…,k}=[k]

BUYERS Alice Bob

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

Task: find partition [k]= S Sc

Max A(S) +B (Sc)

Thm[Nisan,Segal ‘01]: No! Communication exp(k)

Proof:Max A(S) +B (Sc

)=2 iff1-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 bitsPi(x) = Prob [ Alg accesses xi ]

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

1/n A(x)=B(y)=i & xiyiProb[ ] .98/Q(f)

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

Formula SizeCompute f:{0,1}n{0,1}Resources: size, depth

x3

x2

x1 x3

x1

x2

A: x=101 B: y=110

Thm[Karchmer-Wigderson ‘88]:

Pf: find i such that xiyi

Then cc(Pf) = depth (f)

A: f(x)=0 B: f(y)=1

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

VLSI & Distributed Computing

Compute f:{0,1}n{0,1}

Resources: Area, Time

Thm:[Aho,Ullman, Yannakakis ‘83](Area)(Time) cc’(f) (n)

x1

x3x2

f

A B

Projecting Linear ProgramsThm[Khachian ‘80]: Linear Programming P Fact: TSP is a linear programProblem: Exponentially many facets (inequalities)Idea: Write TSP polytope as a projection of

another, with few facetsClaim[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 ProgramsThm[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 fIf LP is the projection of LP’ then#facets (LP’) exp( ncc(hLP) )

/ valid inequalities/ feasible points

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]

x y

z

f(x,y,z)Number onForeheadModel [Chandra, Furst, Lipton ‘83]

The story of

InteractiveProofs

IP[B,GMR]

#PIP [LFKN]IP=PSPACE [S]

MIP[BGKW]

MIP=NEXP[BFL]

PCP [BFLS,FGLSS]

PCP(log n,1)=NP [AS,ALMSS]

InteractiveProofs

OptimizationApprox

ProgramChecking

Property Testing

NP: efficientproofs

Randomized Computation

ProofComplexity

CircuitComplexity

CryptographyZero-Knowledge

#PIP [LFKN]IP=PSPACE [S]

MIP[BGKW]

#PIP [LFKN]IP=PSPACE [S]

MIP[BGKW]

Dist CompInternet

PermanentMIP [N]

Per is RSR[L,BF]

Permanent#P-complete [V]PH-hard [T]Approx [JSV]

Streaming, SublinearAlgorithms

CodingTheory

What is the glue?

Algorithms, likeIterative alg for LPs- Boosting of learning algs- Hard-core sets- On-line routing- Congestion control TCP/IP

- Parallel matching alg

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

Problems, likePermanent- Structural Complexity- Statistical Physics- Comb Optimization- Arithmetic Circuits- Interactive Proofs

Models, likeCommunication

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

What is the glue?

Subject: Computation- Biological processes(DNA,

cell, brain, populations…)- Physical processes (atoms,

weather, galaxies)- Internet, Stock Market- Proofs

Objects, likeExpanders- Data Structures- Derandomization- Networks- Coding Theory- Mathematics

Language, or Level at which we conceptualize

- Asymptotic analysis- Adversaries(worst-case

& amortized analysis)- Generality- Connections/

Reductions

People, likeLes Valiant- Circuit Complexity- Parallel Computation- Learning- Neural Computation- Quantum algorithms

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