210 likes | 430 Views
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.
E N D
Introduction to Complexity Classes Joan Feigenbaum Jan 18, 2007
Computational ComplexityThemes • “Easy” vs. “Hard” • Reductions (Equivalence) • Provability • Randomness
Poly-Time Solvable • Nontrivial Example : Matching
Poly-Time Solvable • Nontrivial Example : Matching
Poly-Time Verifiable • Trivial Example : Hamiltonian Cycle
Poly-Time Verifiable • Trivial Ex. : Hamiltonian Cycle
Is it Easier to Verify a Proof than to Find one? • Fundamental Conjecture of Computational Complexity: PNP
Distinctions • Matching: • HC: Fundamentally Different
Reduction of B to A • If A is “Easy”, then B is, too. A B Algorithm “oracle” “black box”
NP-completeness • P-time reduction • Cook’s theorem If B2NP, then B·P-timeSAT • HC is NP-complete
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…
NP coNP P
Random Poly-time Solvable x2?L YES P-time Algorithm x r NO x2{0,1}n r2{0,1}poly(n)
Probabilistic Classes x2 L “yes” w.p. ¾ x L “no” w.p. 1 x2 L “yes” w.p. 1 x L “no” w.p. ¾ RP coRP (Outdated) Nontrivial Result PRIMES 2 ZPP ´ RPÅ coRP
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?
NP coNP RP coRP ZPP P
Interactive Provability x V [PPT, ] P yes/no
L2IP • x2 L 9P: “yes” w.p. ¾ • x L 8P*: “no” w.p. ¾ Nontrivial Result Interactively Provable Poly-Space Solvable
PSPACE NP coNP RP coRP ZPP P
EXP PSPACE P#P PH iP 2P PH P