Algorithm Design

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Chapter 8 NP and Computational Intractability
of the words. (Multiple copies of words are okay here; so for example, if
the set of refrigerator magnets includes two copies each of C, A, and T, . .....
it would be okay to spell out CAT twice.)
This turned out to be pretty difficult, and it was only later that you
realized a plausible reason for this. Suppose we consider a general version
of the problem of Using Up All the Refrigerator Magnets, where we replace
the English alphabet by an arbitrary collection of syznbols, and we model
Madison’s vocabulary as an arbitrary set of strings over this collection of
symbols. The goal is the same as in the previous paragraph.
Prove that the problem of Using Up All the Refrigerator Magnets is
9.~CPo-~s replete"
ider the following problem. You are managing a communication
network, modeled by a directed graph G = (V, E). There are c users who
are interested in malting use of this network. User i (for each i = t, 2 ..... c)
issues a request to reserve a specific path Pi in G on which to transmit
data.
You are interested in accepting as many of these path requests as
possible, subject to the following restriction: ff you accept both Pi and
then Pi and Pj cannot share any nodes.
Thus, the Path Selection Problem asks: Given a directed graph G =
(V, E), a set of requests P1, P2 ..... pc--each of which must be a path in
G--and a number k, is it possible to select at least k of the paths so that
no two of the selected paths share any nodes?
Prove that Path Selection is NP-complete.
Jl0. Your friends at WebExodus have recently been doing some consullAng
work for companies that maintain large, publicly accessible Web sites-contractual
issues prevent them from saying which ones--and they’ve
come across the following Strategic Advertising Problem.
A company comes to them with the map of a Web site, which we’ll
model as a directed graph G = (V, E). The companY also provides a set of
t trails typically followed by users of the site; we’ll model these trails as
directed paths P1, P2 ..... Pt in the graph G (i.e., each Pi is a path in G).
The companY wants WebExodus to answer the following question
for them: Given G, the paths {P~}, and a number k, is it possible to place
advertisements on at most k of the nodes in G, so that each path P~
includes at least one node containing an advertisement? We’ll call this
the Strategic Advertising Problem, with input G, {P~ : i = 1 ..... t}, and k.
Your friends figure that a good algorithra for this will make them all
rich; tmfortunately, things are never quite this simple.
Exercises
Prove that Strategic Advertising is NP-complete.
(b) Your friends at WebExodus forge ahead and write a pretty fast algorithm
8 that produces yes/no answers to arbitrary instances of the
Strategic Advertising Problem. You may assume that the algorithm
8 is always correct.
Using the algorithm 8 as a black box, design an algorithm that
takes input G, {Pi}, and k as in part (a), and does one of the following
two things:
- Outputs a set of at most k nodes in G so that each path Pi includes
at least one of these nodes, or ’ ~
- Outputs (correctly) that no such set of at most k n6des exists.
Your algorithm should use at most a polynomial number of steps, together
with at most a polynomial number of calls to the .algorithm 8.
11. As some people remember, and many have been told, the idea of hypertext
predates the World Wide Web by decades. Even hypertext fiction is
a relatively old idea: Rather than being constrained by the linearity of
the printed page, you can plot a story that consists of a collection of
interlocked virtual "places" joined by virtual "passages. ’’4 So a piece of
hypertext fiction is really riding on an tmderlying directed graph; to be
concrete (though narrowing the fuli range of what the domain can do),
we’ll model this as follows.
Let’s view the structure of a piece of hypertext fiction as a directed
graph G = (V, E). Each node u e V contains some text; when the reader is
currently at u, he or she can choose to follow any edge out of u; and if the
reader chooses e = (u, v), he or she arrives next at the node v. There is a
start node s ~ V where the reader begins, and an end node t ~ V; when the
reader first reaches t, the story ends. Thus any path from s to t is a valid
plot of the story. Note that, unlike one’s experience using a Web browser,
there is not necessarily a way to go back; once you’ve gone from a to ~,
~ ~ou might not be able to ever return to u.
In this way, the hypertext structure defines a huge number of different
plots on the same underlying content; and the relationships among
all these possibilities can grow very intricate. Here’s a type of problem
one encounters when reasoning about a structure 3ike this. Consider a
piece of hypertext fiction built on a graph G = (V, E) in which there are
certain crucial thematic elements: love, death, war, an intense desire to
major in computer science, and so forth. Each thematic element i is represented
by a set Ti _c V consisting of the nodes in G at which this theme
See, e.g., http://www.eastgate.com.
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