4
The project selection problem
One of the surprising aspects of the maximumflow minimumcut theorem is
that while we started thinking about solving one optimization problem, we
ended up solving two problems for the price of one. We now also have the
possibility of, given a directed graph
G
= (
N, E
), with a specified source node
s
and sink node
t
, where each directed edge has an associated capacity, to
find the cut of minimum capacity. We next give a new optimization problem
that can be formulated as a minimumcut problem.
The project selection problem is as follows.
There is a collection of
projects that one might take on:
P
=
{
P
1
, P
2
, . . . , P
m
}
.
Each project
P
i
has an associated benefit
b
i
,
i
= 1
, . . . , m
. There is also a collection of tools
T
=
{
1
, . . . , n
}
that are needed for doing these projects. Each tool
j
has an
associated cost
c
j
,
j
= 1
, . . . , n
. (One might imagine that the projects are
clients that a management consulting firm might take on, and the tools are
software licenses that are needed for some of these projects.) For each project
P
i
, there is a subset of tools
T
i
⊆ T
that are required to do that project;
once the tool is purchased, it may be used for as many different projects
as required. The aim is to select a subset of the project, and purchase the
requisite tools so as to maximize the net profit associated with these activ
ities (where the net profit is the total benefit accrued minus the total cost
incurred).
We will show how to formulate the project selection problem as a min
imum cut problem. A priori, this seems extremely surprising since there is
no graph as part of the description of the problem, no apparent cuts, and
even we trying to maximize something on the one hand, whereas the mincut
problem is a minimization problem. We will do this formulation by the same
3step process we have repeated for other problems. First, we will show how
to map a project selection input into an input for the minimum cut prob
lem. Then we will demonstrate that feasible solutions to the project selection
problem have a precise correspondence to feasible solutions for the resulting
minimum cut problem. Finally, we will show that optimizing the objective
of the minimum cut problem exactly corresponds to finding the best selec
tion of projects. An example of an input to the project selection problem is
as follows. Suppose that the set of tools is
{
A, B, C, D
}
, and suppose that
the set of projects is
{
P
1
, P
2
, P
3
}
. The requirements for the projects are as
follows: the first requires tools
A
,
B
, and
C
; the second requires
B
and
C
;
and the last requires
C
and
D
. The tool costs are, respectively, 5, 10, 5, and
28
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
15; the project benefits are, respectively, 15, 20, and 10.
At first, it seems that there is no graph structure here at all. However,
it is extremely natural to represent the requirements as a graph, in a way
reminiscent of what we did for the assignment problem earlier. Suppose that
we construct a graph with two sets of nodes: one set corresponding to the
projects, and one set corresponding to the tools. We can draw the first set on
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 SHMOYS/LEWIS
 Optimization, project selection problem, tij xij

Click to edit the document details