1. Computing the Least Common Multiple
2. Counting Paths in a Graph
3. Reduction of Optimization Problems
4. Linear Programming

**Problem Reduction**

Here is my version of a well-known joke about
mathematicians. Professor X, a noted mathematician, noticed that when his wife
wanted to boil water for their tea, she took their kettle from their cupboard,
filled it with water, and put it on the stove. Once, when his wife was away (if
you have to know, she was signing her best-seller in a local bookstore), the
professor had to boil water by himself. He saw that the kettle was sitting on
the kitchen counter. What did Professor X do? He put the kettle in the cupboard
first and then proceeded to follow his wife’s routine.

The way Professor X approached his task is an
example of an important problem-solving strategy called ** problem reduction**. If you
need to solve a problem, reduce it to another problem that you know how to
solve (Figure 6.15).

The joke about the professor notwithstanding,
the idea of problem reduction plays a central role in theoretical computer
science, where it is used to classify problems according to their complexity.
We will touch on this classification in Chapter 11. But the strategy can be
used for actual problem solving, too. The practical difficulty in applying it
lies, of course, in finding a problem to which the problem at hand should be
reduced. Moreover, if we want our efforts to be of practical value, we need our
reduction-based algorithm to be more efficient than solving the original
problem directly.

Note that we have already encountered this
technique earlier in the book. In Section 6.5, for example, we mentioned the
so-called synthetic division done by applying Horner’s rule for polynomial
evaluation. In Section 5.5, we used the following fact from analytical
geometry: if *p*_{1}*(x*_{1}*, y*_{1}*), p*_{2}*(x*_{2}*, y*_{2}** ),** and

is positive if and only if the point *p*_{3} is to
the left of the directed line *p*_{1}*p*_{2} through points *p*_{1} and *p*_{2}. In other words, we reduced a geometric question about the
relative locations of three points to a question about the sign of a
determinant. In fact, the entire idea of analytical geometry is based on
reducing geometric problems to algebraic ones. And the vast majority of
geometric algorithms take advantage of this historic insight by Rene´ Descartes
(1596–1650). In this section, we give a few more examples of algorithms based
on the strategy of problem reduction.

**Computing the Least Common
Multiple**

Recall that the ** least common multiple** of
two positive integers

lcm** (**24

As a computational procedure, this algorithm
has the same drawbacks as the middle-school algorithm for computing the
greatest common divisor discussed in Section 1.1: it is inefficient and
requires a list of consecutive primes.

A much more efficient algorithm for computing
the least common multiple can be devised by using problem reduction. After all,
there is a very efficient algorithm (Euclid’s algorithm) for finding the
greatest common divisor, which is a product of all the common prime factors of ** m** and

where gcd** (m, n)** can be computed very efficiently by Euclid’s
algorithm.

**Counting Paths in a Graph**

As our next example, we consider the problem of
counting paths between two vertices in a graph. It is not difficult to prove by
mathematical induction that the number of different paths of length ** k >** 0 from the

As a specific example, consider the graph of
Figure 6.16. Its adjacency matrix ** A **and its square

paths of length 2 that start and end at vertex ** a** (

**Reduction of Optimization
Problems**

Our next example deals with solving
optimization problems. If a problem asks to find a maximum of some function, it
is said to be a ** maximization problem**; if it asks to find a function’s minimum,
it is called a

min ** f (x)** = − max[−

In other words, to minimize a function, we can maximize
its negative instead and, to get a correct minimal value of the function
itself, change the sign of the answer. This property is illustrated for a
function of one real variable in Figure 6.17.

Of course, the formula

max ** f (x)** = − min[−

is valid as well; it shows how a maximization
problem can be reduced to an equivalent minimization problem.

This relationship between minimization and
maximization problems is very general: it holds for functions defined on any
domain ** D**. In particular, we can

apply it to functions of several variables
subject to additional constraints. A very important class of such problems is
introduced below in this section.

Now that we are on the topic of function
optimization, it is worth pointing out that the standard calculus procedure for
finding extremum points of a function is, in fact, also based on problem
reduction. Indeed, it suggests finding the function’s derivative ** f (x)** and then solving the equation

**Linear Programming**

Many problems of optimal decision making can be
reduced to an instance of the ** linear programming** problem—a problem
of optimizing a linear function of several variables subject to constraints in
the form of linear equations and linear inequalities.

**EXAMPLE 1 **Consider a university endowment that needs to
invest $100 million.** **This sum has to be split between three types of investments:
stocks, bonds, and cash. The endowment managers expect an annual return of 10%,
7%, and 3% for their stock, bond, and cash investments, respectively. Since
stocks are more risky than bonds, the endowment rules require the amount
invested in stocks to be no more than one-third of the moneys invested in
bonds. In addition, at least 25% of the total amount invested in stocks and
bonds must be invested in cash. How should the managers invest the money to
maximize the return?

Let us create a mathematical model of this
problem. Let ** x, y,** and

Although this example is both small and simple,
it does show how a problem of optimal decision making can be reduced to an
instance of the general linear programming problem

(The last group of constraints—called the
nonnegativity constraints—are, strictly speaking, unnecessary because they are
special cases of more general constraints *a _{i}*

Linear programming has proved to be flexible
enough to model a wide variety of important applications, such as airline crew
scheduling, transportation and communication network planning, oil exploration
and refining, and industrial production optimization. In fact, linear
programming is considered by many as one of the most important achievements in
the history of applied mathematics.

The classic algorithm for this problem is
called the ** simplex method** (Sec-tion 10.1). It was discovered by the U.S.
mathematician George Dantzig in the 1940s [Dan63]. Although the worst-case
efficiency of this algorithm is known to be exponential, it performs very well
on typical inputs. Moreover, a more recent al-gorithm by Narendra Karmarkar
[Kar84] not only has a proven polynomial worst-case efficiency but has also
performed competitively with the simplex method in empirical tests.

It is important to stress, however, that the
simplex method and Karmarkar’s algorithm can successfully handle only linear
programming problems that do not limit its variables to integer values. When
variables of a linear programming problem are required to be integers, the
linear programming problem is said to be an ** integer linear programming**
problem. Except for some special cases (e.g., the assignment problem and the
problems discussed in Sections 10.2–10.4), integer linear programming problems
are much more difficult. There is no known polynomial-time algorithm for
solving an arbitrary instance of the general integer linear programming problem
and, as we see in Chapter 11, such an algorithm quite possibly does not exist. Other
approaches such as the branch-and-bound technique discussed in Section 12.2 are
typically used for solving integer linear programming problems.

**EXAMPLE 2 **Let us see how the knapsack problem can be
reduced to a linear** **programming problem. Recall from Section 3.4 that the knapsack
problem can be posed as follows. Given a knapsack of capacity ** W** and

EXAMPLE 2 Let us see how the knapsack problem
can be reduced to a linear programming problem. Recall from Section 3.4 that
the knapsack problem can be posed as follows. Given a knapsack of capacity W
and n items of weights w1, . . . , wn and values v1, . . . , vn, find the most
valuable subset of the items that fits into the knapsack. We consider first the
continuous (or fractional) version of the problem, in which any fraction of any
item given can be taken into the knapsack. Let xj , j = 1, . . . , n, be a
variable representing a fraction of item j taken into the knapsack. Obviously,
xj must satisfy the inequality 0 ≤ xj ≤ 1. Then the total

This seemingly minor modification makes a
drastic difference for the com-plexity of this and similar problems constrained
to take only discrete values in their potential ranges. Despite the fact that
the 0-1 version might seem to be eas-ier because it can ignore any subset of
the continuous version that has a fractional value of an item, the 0-1 version
is, in fact, much more complicated than its con-tinuous counterpart. The reader
interested in specific algorithms for solving this problem will find a wealth
of literature on the subject, including the monographs [Mar90] and [Kel04].

**Reduction to Graph Problems**

As we pointed out in Section 1.3, many problems
can be solved by a reduction to one of the standard graph problems. This is
true, in particular, for a variety of puzzles and games. In these applications,
vertices of a graph typically represent possible states of the problem in
question, and edges indicate permitted transi-tions among such states. One of
the graph’s vertices represents an initial state and another represents a goal
state of the problem. (There might be several vertices of the latter kind.)
Such a graph is called a ** state-space graph**. Thus, the
transfor-mation just described reduces the problem to the question about a path
from the initial-state vertex to a goal-state vertex.

**EXAMPLE **Let us revisit the classic river-crossing
puzzle that was included in** **the exercises for Section 1.2. A peasant finds himself on a river
bank with a wolf, a goat, and a head of cabbage. He needs to transport all
three to the other side of the river in his boat. However, the boat has room
only for the peasant himself and one other item (either the wolf, the goat, or
the cabbage). In his absence, the wolf would eat the goat, and the goat would
eat the cabbage. Find a way for the peasant to solve his problem or prove that
it has no solution.

The state-space graph for this problem is given
in Figure 6.18. Its vertices are labeled to indicate the states they represent:
P, w, g, c stand for the peasant, the wolf, the goat, and the cabbage,
respectively; the two bars | | denote the river; for convenience, we also
label the edges by indicating the boat’s occupants for each crossing. In terms
of this graph, we are interested in finding a path from the initial-state
vertex labeled Pwgc| | to the final-state vertex labeled | |Pwgc.

It is easy to see that there exist two distinct
simple paths from the initial-state vertex to the final state vertex (what are
they?). If we find them by applying breadth-first search, we get a formal proof
that these paths have the smallest number of edges possible. Hence, this puzzle
has two solutions requiring seven river crossings, which is the minimum number
of crossings needed.

Our success in solving this simple puzzle
should not lead you to believe that generating and investigating state-space
graphs is always a straightforward task. To get a better appreciation of them,
consult books on artificial intelligence (AI), the branch of computer science
in which state-space graphs are a principal subject.

In this book, we deal with an important special
case of state-space graphs in Sections 12.1 and 12.2.

**Exercises 6.6**

**1.
a. **Prove the equality

that underlies the algorithm for computing lcm*(m, n).*

**
**Euclid’s
algorithm is known to be in ** O(**log

**
**You are
given a list of numbers for which you need to construct a min-heap. (A min-heap
is a complete binary tree in which every key is less than or equal to the keys
in its children.) How would you use an algorithm for constructing a max-heap (a
heap as defined in Section 6.4) to construct a min-heap?

**
**Prove
that the number of different paths of length ** k >** 0 from the

**
****a. **Design an algorithm with a time efficiency better than cubic for
checking** **whether a graph with ** n** vertices contains a cycle of length 3 [Man89].

**
**Consider
the following algorithm for the same problem. Starting at an arbi-trary vertex,
traverse the graph by depth-first search and check whether its depth-first
search forest has a vertex with a back edge leading to its grand-parent. If it
does, the graph contains a triangle; if it does not, the graph does not contain
a triangle as its subgraph. Is this algorithm correct?

**
**Given ** n >** 3 points

**
**Consider
the problem of finding, for a given positive integer ** n,** the pair of integers whose sum is

7. The assignment problem introduced in Section
3.4 can be stated as follows: There are ** n** people who need to be assigned to execute

**8. **Solve the instance of the linear programming
problem given in Section 6.6:

** **9. **
**The
graph-coloring problem is usually stated as the vertex-coloring prob-lem:
Assign the smallest number of colors to vertices of a given graph so that no
two adjacent vertices are the same color. Consider the ** edge-coloring** problem:
Assign the smallest number of colors possible to edges of a given graph so that
no two edges with the same endpoint are the same color. Ex-plain how the
edge-coloring problem can be reduced to a vertex-coloring problem.

** **11. **
***Jealous husbands *There are*
**n** *≥* *2 married couples who need to cross a* *river. They have a boat that can hold
no more than two people at a time. To complicate matters, all the husbands are
jealous and will not agree on any crossing procedure that would put a wife on
the same bank of the river with another woman’s husband without the wife’s
husband being there too, even if there are other people on the same bank. Can
they cross the river under such constraints?

**
**Solve
the problem for ** n** = 2

**
**Solve
the problem for ** n** = 3, which is the classical version of this
problem.

**
**Does the
problem have a solution for ** n** ≥ 4? If it does, indicate how many river
crossings it will take; if it does not, explain why.

12. *Double-n
dominoes *Dominoes are small
rectangular tiles with dots called* *spots
or pips embossed at both halves of the tiles. A standard “double-six” domino
set has 28 tiles: one for each unordered pair of integers from (0** ,** 0) to (6

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