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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. 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)

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  1. Depth through Breadth(or, why should we go to talks in other areas) Avi Wigderson IAS, Princeton

  2. 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]

  3. 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)

  4. 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)

  5. 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!)

  6. (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

  7. 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

  8. 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)

  9. 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!

  10. 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

  11. 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]

  12. The story of Interactive Proofs

  13. 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]

  14. 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

  15. 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

  16. 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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