depth through breadth (or, why should we go to talks in other areas) avi wigderson ias, princeton
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Depth through Breadth(or, why should we go to
talks in other areas)
Avi WigdersonIAS, Princeton
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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
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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)
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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)
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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!)
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(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
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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
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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
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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!
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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
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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]
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The story of
InteractiveProofs
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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
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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
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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
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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