Lecture plan for Wednesday Nov. 5, 9.15 - 10.20:

Satisfiability
Maximum SAT
Maximum 3-SAT
Approximating Maximum 3-SAT
A new class of algorithms: Randomized algorithms
  Monte Carlo
Random Variable 
Expected value
For Maximum 
Linearity of expectation
For Maximum 3-SAT: b=1/2, E(Z(F,b)) = 7/8.

Derandomization: use recurrence relation:
 E(Z(F,b)) = b*E(Z(F[x=1],b)) + (1-b)*E(Z(F[x=0],b))

In Dynamic Programming examples you have seen so far,
the recurrence was in terms of the optimal solution.
Here it is in terms of the expected value.

Derive algorithm from recurrence relation.
Original problem is split into two overlapping subproblems.

Gives efficient algorithm for Maximum 3-SAT guaranteeing 7/8.

What if it is not 3-SAT: need to use a bent coin: compute maximum bias.
But algorithm stays the same.

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Connection to Dynamic Programming:
Weighted Interval Scheduling
Recurrence Relation:
Opt(j)=max(v_j + Opt(p(j)), Opt(j-1))

Compute-Opt(j) // inefficient
  if j=0 then return 0 else
    return max(v_j+Compute-Opt(p(j)), Compute-Opt(j-1))

ComputeA(F,b) // returns an assignment satisfying E(Z(F,b))
  if F has no variables return {} else
    choose a variable x; 
    pos = E(Z(F[x=1],b));
    neg = E(Z(F[x=0],b));
    if pos > neg return (ComputeA(F[x=1],b) union 1) else
                 return (ComputeA(F[x=0],b) union 0);

Differences:
Recurrence relation is in terms of expected value,
not the optimum.
The expected value we can compute in closed form in polynomial time.

WIS   CSP
j     number of variables
2 calls  1 call