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COMP SCI 731 - Lecture 3 - Tuesday, Nov 6, 2000
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Today:  o decision vs. search:
        o The language framework


Decision vs. Search
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Describe 3SAT and CLIQUE *optimization* problems.
Explain why they are at least as hard as the corresponding decision problems.
Now prove that the CLIQUE decision problem is at least as hard as the
CLIQUE optimization problem:


int max-clique(G) // G=(V,E) where |V|=n
begin
  for i= n downto 1 do
    if CLIQUE(G,i) then return i
end 

find-biggest-clique (G) 
begin
  k = max-clique(G)
  for i = 1 to n do
     if CLIQUE(G \ {i},k) then G = G \ {k}
  return V(G)
end


How many calls to CLIQUE does the above use?   n + lg(n). You can reduce
to n (or to n - c lg n) by stopping when the graph becomes small and looking
for the biggest clique by the "naive" algorithm.  

Now, for homework, do the same thing for 3SAT.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Problem 1.  Suppose I give you black box that solves the
            3SAT decision problem.  Given an algorithm (that uses this
            black box) to find a satisfying formula to an input formula
            phi.    How many calls to the black box do you need?
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++




The language framework
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A language is a set of strings.
Decision problems and languages are in one-to-one correspondence:
Talk about decision verses search.

Decision problem Pi:

   Given a string x, is x a Yes-instance of Pi ?

Language L_Pi

   {x: x is a Yes-instance of Pi}

People often speak in terms of this language framework.  
You should be comfortable with it.