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Introduction to Complexity Classes

Introduction to Complexity Classes. Joan Feigenbaum Jan 18, 2007. Computational Complexity Themes. “Easy” vs. “Hard” Reductions (Equivalence) Provability Randomness. Poly-Time Solvable. Nontrivial Example : Matching. Poly-Time Solvable. Nontrivial Example : Matching.

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Introduction to Complexity Classes

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  1. Introduction to Complexity Classes Joan Feigenbaum Jan 18, 2007

  2. Computational ComplexityThemes • “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. Distinctions • Matching: • HC: Fundamentally Different

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

  10. NP-completeness • P-time reduction • Cook’s theorem If B2NP, then B·P-timeSAT • 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. NP coNP P

  13. Random Poly-time Solvable x2?L YES P-time Algorithm x r NO x2{0,1}n r2{0,1}poly(n)

  14. Probabilistic Classes x2 L  “yes” w.p. ¾ x L  “no” w.p. 1 x2 L  “yes” w.p. 1 x L  “no” w.p. ¾ RP coRP (Outdated) Nontrivial Result PRIMES 2 ZPP ´ RPÅ coRP

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

  16. NP coNP RP coRP ZPP P

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

  18. L2IP • x2 L 9P: “yes” w.p. ¾ • x L 8P*: “no” w.p. ¾ Nontrivial Result Interactively Provable Poly-Space Solvable

  19. PSPACE NP coNP RP coRP ZPP P

  20. EXP PSPACE P#P PH iP 2P PH P

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