Solutions to the COMP360 Winter 2006 Midterm
1. Selling cars (10 points)
(a)
The objects to be chosen are customercar pairs. For example, selling Car B to Curly would
be represented by the pair (Curly, Car B). There are thus
N
2
objects to choose from. The weight
of such a pair, of course, is the money earned by the salesman. Most students gave one of two
possible definitions for an admissible solution. Version 1 is: An admissible solution is any set
of objects such that no person and no car appears more than once. Version 2 is: An admissible
solution is a set of precisely
N
objects such that no person and no car appears more than once. (In
other words, only complete matchings between people and cars are allowed as solutions. This is
reasonable because a solution of size less than
N
can not be optimal, assuming strictly positive car
costs.)
(b)
The two main properties are Heredity, or Inheritance, and the Exchange Property. The Heredity
property states that if
S
is an admissible solution and
S
′
⊂
S
, then
S
′
is an admissible solution.
The Exchange Property states that if
S
and
S
′
are two admissible solutions and

S

<

S
′

, then
there is some object
x
∈
S
′
such that
S
∪ {
x
}
is an admissible solution.
(c)
Version 1 satisfies the inheritance property trivially. It does not satisfy Exchange. For example,
let
S
′
=
{
(Larry, Car A), (Moe, Car B)
}
and
S
=
{
(Moe, Car A)
}
. We have

S

<

S
′

, but neither
object from
S
′
can be added to
S
. Version 2 does not satisfy inheritance, because any strict subset
of an admissible solution is not of size
N
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 Winter '03
 Perkins
 Dynamic Programming, Shortest path problem, Iter 3 Iter 4 Iter, admissible solution

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