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COMP SCI 731 - Lecture 5 - Tuesday, Nov 13, 2000
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Today: o P and NP
       o Definitions of reductions and NP-completeness
       o Another reduction: HC < TSP

Announcemnt: quiz next Tuesday



P and NP
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Make a broad distinction: "efficient" verses "inefficient" corresponding to P and not P.

Describe NP informally.  

Go through example problems:

 CONNECTIVITY
 3SAT
 CLIQUE
 SUBSET SUM
 PRIMES
 
Which are in P and which are in NP?

Formally define P and NP.
 
P = { L: L is a language and there exists a TM M s.t.
          * x in L     ==> M(x) says YES
          * x not in L ==> M(x) says NO
          * For some polynomial poly, for all x, #steps_M(x) <= poly(|x|)

NP = { L: L is a language and there exists a TM V (the "verifier") such that
          * x in L ==> there is some c ("the certificate") s.t. M(x,c)=YES
          * x not in L ==> for all c M(x,c)=NO
          * For some polynomial poly, for all x, #steps_M(x,c) <= poly(|x|) }

Reductions
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Turing-reductions:

  Let A and B be problems.  We say that A \redt B if there is a polynomial
  time algorithm M, having oracle access to B, that solves A.


Many-one poly-time reductions.

  Let A and B be languages (decision questions).  
  We say that A \redp B if there is a polynomial-time computable f such that 
  x in A iff f(x) in B.


We will use the latter type of reduction.

* Prove transitivity of \redp 


Definition [Cook-Levin]
  ------------------------------------
  A language A is NP-complete if 
  (1) A is in NP
  (2) If L is in NP then L \redp A
  ------------------------------------

Strategy for showing A is NP-complete

  (1) Show that A is in NP.
  (2) Show that, for some NP-complete problem L,   L\redp A.

Explain why this works.

Let us SUPPOSE that 3SAT is NP-complete.
Then we can CONCLUDE that CLIQUE is NP-Complete, because we've show
that 3SAT redp CLIQUE.


A reduction
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HC < TSP