Algorithm Design

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Chapter 8 NP and Computational Intractability
party views the set of goods they’re getting as more valuable than the set
of goods they’re giving in return. Historically, societies tend to move from
barter-based to money-based economies; thus various online systems
that have been experimenting with barter can be viewed as intentional
attempts to regress to this earlier form of economic interaction. In doing
this, they’ve rediscovered some of the inherent difficulties with barter
relative to money-based systems. One such difficulty is the complexity
of identifying opportunities for trading, even when these opportunities
exist. To model this complexity, we need a notion that each person assigns
a value to each object in the world, indicating how much this object would
be worth to them. Thus we assume there is a set of n people p~ ..... P~,
and a set of m distinct objects al ..... am. Each object is owned by one
of the people. Now each person Pi has a valuation function v~, defined so
that vdaj) is a nonnegative number that specifies how much object aj is
worth to p~--the larger the number, the more valuable the object is to the
person. Note that everyone assigns a valuation to each object, including
the ones they don’t currently possess, and different people can assign
very different valuations to the same object.
A two-person trade is possible in a system like this when there are
people p~ and pj, and subsets of objects A~ and A1 possessed by p~ and pi,
respectively, so that each person would prefer the objects in the subset
they don’t currently have. More precisely,
¯ pi’s total valuation for the objects in A1 exceeds his or her total
valuation for the objects in &, and
. pis total valuation for the objects in A~ exceeds his or her total
valuation for the objects in Aj.
(Note that A~ doesn’t have to be all the objects possessed b y p~ (and
likewise for Ai); Ai and A1 can be arbitrary subsets of their possessions
that meet these criteria.)
Suppose you are given an instance of a barter economy, specified
by the above data on people’s valuations for objects. (To prevent problems
with representing real numbers, we’ll assume that each person’s
valuation for each object is a natural number.) Prove that the problem of
determining whether a two-person trade is possible is NP-complete.
34. in the 1970s, researchers including Mark Granovetter and Thomas
Schelling in the mathematical social sciences began trying to develop
models of certain kinds of collective human behaviors: Why do particular
fads catch on while others die out? Why do particular technological
innovations achieve widespread adoption, while others remain focused
Exercises
on a sma~ group of users? What are the dynamics by which rioting and
looting behavior sometimes (but only rarely) emerges from a crowd of
angry people? They proposed that these are all examples of cascade
processes, in which an individual’s behavior is highly influenced by the
behaviors of his or her friends, and so ff a few individuals instigate the
process, it can spread to more and more people and eventually have a
very wide impact. We can think of this process as being like the spread
of an illness, or a rumor, jumping from person to person.
The mos~ basic version of their models is the following. There is some
underlying behavior (e.g., playing ice hockey, owning a cell phone, taking
part in a riot), and at any point in time each person is either an adopter of
the behavior or a nonadopter. We represent the population by a directed
graph G = (V, E) in which the nodes correspond to people and there is
an edge (v, w) ff person ~ has influence over the behavior of person w: If
person v adopts the behavior, then this helps induce person u; to adopt
it as well. Each person w also has a given threshold O(w) ~ [0, 1], and this
has the following meaning: At any time when at least a O(w) fraction of
the nodes with edges to tv are adopters of the behavior, the node tv will
become an adopter as well.
Note that nodes with lower thresholds are more easily convinced
to adopt the behavior, while nodes with higher thresholds are more
conservative. A node w with threshold O(w) = 0 will adopt the behavior
immediately, with no inducement from friends. Finally, we need a convention
about nodes with no incoming edges: We will say that they become
adopters ff O(w) = 0, and cannot become adopters if they have any larger
threshold.
Given an instance of this model, we can simulate the spread of the
behavior as follows.
Initially, set all nodes w with 0(w)=0 to be adopters
(All other nodes start out as nonadopters)
Until there is no change in the set of adopters:
For each nonadopter w simultaneously:
If at least a O(w) fraction of nodes w~ith edges to w are
adopters then
~3 becomes an adopter
Endif
End~or
End
Output the final set of adopters
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