CS365/Schedule/2004-04-15

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Poly-time reductions, where one shows that A ≤_P B by showing a poly-time f such that A(x) = B(f(x)).
NP-hard problems: B is NP-hard if A ≤_P B for any A in NP.
B is NP-complete if it is NP-hard and if it is in NP.
Proving NP-completeness:
- Basic idea: if A is NP-hard and A ≤_P B, then B is also NP-hard. Consequences:
  - Showing that some NP-hard A is in P implies P=NP.
  - If P≠NP, then every NP-hard A is not in P.
  - NP-complete problems are the test cases for P=?NP; either they are all in P (and P=NP), or they are all not in P (and P≠NP).
- Cook's Theorem: SAT is NP-complete.
- Other reductions: SAT ≤_P 3SAT ≤_P CLIQUE
- Reductions we didn't get to: CLIQUE ≤_P INDEPENDENT SET ≤_P VERTEX COVER ≤_P SUBSET SUM ≤_P PARTITION.

See PvsNp and SubsetSumReduction for some of the missing ones and examples of other NP-complete problems. See GareyAndJohnson for many many such problems.