1 Introduction to Complexity Classes Joan Feigenbaum Jan 18, 2007

2 Computational Complexity Themes “Easy” vs. “Hard” Reductions (Equivalence) Provability Randomness

3 Poly-Time Solvable Nontrivial Example : Matching

4 Poly-Time Solvable Nontrivial Example : Matching

5 Poly-Time Verifiable Trivial Example : Hamiltonian Cycle

6 Poly-Time Verifiable Trivial Ex. : Hamiltonian Cycle

7 Is it Easier to Verify a Proof than to Find one? Fundamental Conjecture of Computational Complexity: P  NP

8 Matching: HC: Fundamentally Different Distinctions

9 Reduction of B to A If A is “Easy”, then B is, too. B Algorithm A “oracle” “black box”

10 NP-completeness P-time reduction Cook’s theorem If B 2 NP, then B · P-time SAT HC is NP-complete

11 Equivalence NP-complete probs. are an Equivalence Class under P-time reductions. 10k’s problems Diverse fields Math, CS, Engineering, Economics, Physical Sci., Geography, Politics…

12 NPcoNP P

13 Random Poly-time Solvable x2?Lx2?L P-time Algorithm x r YES NO x 2 {0,1} n r 2 {0,1} poly(n)

14 Probabilistic Classes x 2 L  “yes” w.p. ¾ x  L  “no” w.p. 1 x 2 L  “yes” w.p. 1 x  L  “no” w.p. ¾ RP coRP (Outdated) Nontrivial Result PRIMES 2 ZPP ´ RP Å coRP

15 Two-sided Error BPP x  L  “yes”w.p. ¾ x  L  “no” w.p. ¾ Question to Audience: BPP set not known to be in RP or coRP?

16 RPcoRP ZPP NPcoNP P

17 Interactive Provability P V [PPT,  ] x yes/no

18 L 2 IP x 2 L  9 P: “yes” w.p. ¾ x  L  8 P*: “no” w.p. ¾ Nontrivial Result Interactively Provable Poly-Space Solvable

19 PSPACE RPcoRP ZPP NPcoNP P

20 PSPACE EXP P #P PH iPiP 2P2P P