Algorithm Design

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Chapter 7 Network Flow
Lower bound of 2 ~ ~
(a)
b~oiuminating a lower ~
nd from an edge.)
Figure 7.15 (a) An instance of the Circulation Problem with lower bounds: Numbers
inside the nodes are demands, and numbers labeling the edges are capacities. We also
assign a lower bound of 2 to one of the edges. (b) The result of reducing this instance
to an equivalent instance of the Circulation Problem without lower bounds.
(7.52) There is a feasible circulation in G if and only if there is a feasible
circulation in G’. If all demands, capacities, and lower bounds in G are integers,
and there is a feasible circulation, then there is a feasible circulation that is
integer-valued.
Proof. First suppose there is a circulation f’ in G’. Define a circulation f in G
by f(e) = f’(e) + ge- Then f satisfies the capacity conditions in G, and
fin(v) - / °ut(v) = E (ge + f’(e)) - (ge + f’(e)) = Lv + (d~ - Lv) = d~,
e into v
e out of v
so it satisfies the demand conditions in G as well.
Conversely, suppose there is a circulation f in G, and define a circulation
f’ in G’ by f’(e) = f(e) - ~e. Then f’ satisfies the capacity conditions in G’, and
(f’)in(v) - (f ’)°ut(v) = E (f(e) - g.~) - if(e) - ~’e) =
e into v
e out of v
so it satisfies the demand conditions in G’ as well.
7.8 Survey Design
Many problems that arise in applications can, in fact, be solved efficiently by
a reduction to Maximum Flow, but it is often difficult to discover when such
a reduction is possible. In the next few sections, we give several paradigmatic
examples of such problems. The goal is to indicate what such reductions tend
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7.8 Survey Design
to look like and to illustrate some of the most common uses of flows and cuts
in the design of efficient combinatorial algorithms. One point that will emerge
is the following: Sometimes the solution one wants involves the computation
of a maximum flow, and sometimes it involves the computation of a minimum
cut; both flows and cuts are very useful algorithmic tools.
We begin with a basic application that we call survey design, a simple
version of a task faced by many companies wanting to measure customer
satisfaction. More generally, the problem illustrates how the construction used
to solve the Bipartite Matching Problem arises naturally in any setting where
we want to carefully balance decisions across a set of options--in this case,
designing questionnaires by balancing relevant questions across a population
of consumers.
~ The Problem
A major issue in the burgeoning field of data mining is the study of consumer
preference patterns. Consider a company that sells k products and has a
database containing the purchase histories of a large number of customers.
(Those of you with "Shopper’s Club" cards may be able to guess how this data
gets collected.) The company wishes to conduct a survey, sending customized
questionnaires to a particular group of n of its customers, to try determining
which products people like overall.
Here are the guidelines for designing the survey.
Each customer wil! receive questions about a certain subset of the
products.
A customer can only be asked about products that he or she has purchased.
To make each questionnaire informative, but not too long so as to discourage
participation, each customer i should be asked about a number
of products between q and c[.
Finally, to collect sufficient data about each product, there must be
between pj and pj distinct customers asked about each product j.
More formally, the input to the Survey Design Problem consists of a bipartite
graph G whose nodes are the customers and the products,’and there is an edge
between customer i and product j if he or she has ever purchased product j.
Further, for each customer i = ! ..... n, we have limits ci < c[ on the number
of products he or she can be asked about; for each product j = 1 ..... k, we
have limits pj