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Depth through Breadth (or, why should we go to talks in other areas) Avi Wigderson IAS, 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 Computing Yes! e.g Communication Complexity [Yao] x Alice y Bob

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Combinatorial Auctions Seller: Goods {1,2,3,…,k}=[k] BUYERS B 1 B 2 B 3 …… B n 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]= S 1 S 2 … S n Max B 1 (S 1 ) +B 2 (S 2 ) +…+ B n (S n ) Basic Question: Can they find it efficiently Polytime (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]= S A S B Max A(S A ) +B (S B ) Basic Question: Can they find it efficiently Polytime (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 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 S c Max A(S) +B (S c ) Thm[Nisan,Segal ‘01]: No! Communication exp(k) Proof: Max A(S) +B (S c )=2 iff 1-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 bits P i (x) = Prob [ Alg accesses x i ] Thm[Ambainis ‘01]: A: f(x)=0 B: f(y)=1 1/n A(x)=B(y)=i & x i y i Prob[ ] .98/ Q(f) f=OR [Grover search] x=0, y=e j for random j

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Formula Size Compute f:{0,1} n {0,1} Resources: size, depth x3x3 x2x2 x1x1 x3x3 x1x1 x2x2 A: x=101 B: y=110 Thm[Karchmer-Wigderson ‘88]: P f : find i such that x i y i Then cc(P f ) = 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) x1x1 x3x3 x2x2 f AB

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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 LP 1 (with n 8 vars) Ref 1 : Bug in LP 1 Claim[Swart ‘87]: P=NP via LP 2 (with n 10 vars) Ref 2 : Bug in LP 2 Thm[Yannakakis ‘88]: Swart’s approach must fail!

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Projecting Linear Programs Thm[Yannakakis]: Let LP be any program. Set up the following CC problem h LP A’s inputs: facets of LP B’s inputs: vertices of LP h LP (f,v)=1 iff v is not on f h LP (f,v)=0 iff v is on f If LP is the projection of LP’ then #facets (LP’) exp( ncc(h LP ) ) / 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] ACC 0 NC 1 ? [Yao] Space pseudorandom gen [Babai, Nisan, Szegedy] x y z f(x,y,z) Number on Forehead Model [Chandra, Furst, Lipton ‘83]

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The story of Interactive Proofs

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IP [B,GMR] #P IP [LFKN] IP=PSPACE [S] MIP [BGKW] MIP=NEXP [BFL] PCP [BFLS,FGLSS] PCP(log n,1)=NP [AS,ALMSS] Interactive Proofs Optimization Approx Program Checking Property Testing NP: efficient proofs Randomized Computation Proof Complexity Circuit Complexity Cryptography Zero-Knowledge #P IP [LFKN] IP=PSPACE [S] MIP [BGKW] #P IP [LFKN] IP=PSPACE [S] MIP [BGKW] Dist Comp Internet Permanent MIP [N] Per is RSR [L,BF] Permanent #P-complete [V] PH-hard [T] Approx [JSV] Streaming, Sublinear Algorithms Coding Theory

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What is the glue? Algorithms, like Iterative alg for LPs -Boosting of learning algs -Hard-core sets -On-line routing -Congestion control TCP/IP -Parallel matching alg 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 Models, like Communication 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, 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 People, like Les 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

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