1
Total points possible
: 20
1.
Satisfiability Formulation
:
Ann, Brad, Cindy, Dan, Eugene and Frank have to get back to their apartment from a
club. They have exactly one car, and not all can fit into the car. Anyone not in the car will
have to get a ride from the bartender who will not leave for another hour. After an
inebriated argument they decide on the following:
(a)
Only Eugene and Dan can drive, so at least one of them should be in the car
(b)
At least one of the women should be in the car
(c)
If Ann is not in the car then Brad and Frank must accompany her since the
bartender has been hitting on her
(d)
If Cindy is in the car then Frank cannot be in the car.
If Cindy is not in the car
then Eugene must go with her
Consider the problem of assigning people to go in the car given the constraints above:
(i)
(1 point)
How many decision variables should this problem have in order to
be formulated as a 2SAT problem? What is the total number of possible
assignments?
This problem should have 6 decision variables in order to be formulated as a 2SAT
problem. This 2SAT problem is formulated so that it can be coded into a computer as a
cost function in the form
±²³´ µ ¶±²·´ µ ¸
. The Boolean operator inside the
brackets should be an “or” operator, and the Boolean operator outside of the brackets
should be an “and” operator. Each variable represents the states of one person, i.e., in or
not in the car. Then, the total number of possible assignments is
¹
º»¼½¾¿ÀÁÂÀÃÄ¿ÅÄ½Æ¾Ç
È ¹
É
È ÊË
Note that this means we can have the assignment [1 1 1 1 1 1] (all people are in the car)
or [0 0 0 0 0 0] (no one is in the car). The computer will not output these as feasible
solutions because of the constraints, but it should still be included in the total number of
possible assignments.
Cannot have combinations (i.e. 6C1*5C1*4C1*3C1*2C1 = 6!) because the SAT problem
reduces the number of possible combinations.
(ii)
(4 points, one point for each of the 4 conditions satisfied above)
Identify
appropriate clauses using only 2 literals in each to reflect their requirements.
Use the first letter of the name to represent each person in the clauses. Give
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 Fall '11
 Shoemaker
 Optimization, Cindy, TABU Search, V*, Tabu.m

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