Algorithm Design

434
Chapter 7 Network Flow
31.
moving to the new system are in fact negative: If bi < 0, then it is preferable
(by an amount quantified in hi) to keep i on the old system. Again, give
a polynomial-time algorithm to find a set $ ~_ {1, 2 ..... n} for which the
sum of the benefits mlnus the expenses of moving the applications in S
to the Sew system is maximized.
Some of your friends are interning at the small high-tech company Web-
Exodus. A running joke among the employees there is that the back room
has less space devoted to high-end servers than it does to empty boxes
of computer equipment, piled up in case something needs to be shipped
back to the supplier for maintainence.
A few days ago, a large shipment of computer monitors arrived, each
in its own large box; and since there are many different kinds of monitors
in the shipment, the boxes do not all have the same dimensions. A bunch
of people spent some time in the morning trying to figure out how to
store all these things, realizing of course that less space would be tkken
up If some of the boxes could be nested inside others.
Suppose each box i is a rectangnlar parallelepiped with side lengths
equal to (i 1, iz, i3); and suppose each side length is strictly between half a
meter and one meter. Geometrically, you know what it means for one box
to nest inside another: It’s possible if you can rotate the smaller so that
it fits inside the larger in each dimension. Formally, we can say that box
i with dimensions (il, i2, i3) nests inside boxj with dimensions (Jl,J2,J3) If
there is a permutation a, b, c of the dimensions {1, 2, 3} so that in