CHAPTER SIX
The
Probabilistic
Method
"lhe probab,ilisti, m"thod
is away of proving
the existence of objects. The underly
ing principle is simple: to prove
the existence of an object
with certain properties, we
demonstrate a sample space
of objects in which
the probability is positive that a ran
domly selected object has the
required properties.
If
the probability of selecting an
object with the required properties
is positive, then the sample space must
contain such
an object, and therefore such an
object exists. For example,
if there is
a positive proba
bility of winning
a
milliondollar pize in
a raffle, then there must
be at least one raffle
ticket that wins that
Prize.
Although the basic principle of
the probabilistic method
is simple, its application to
specific problems often involves sophisticated combinatorial arguments. In this chap
ter we study a number of techniques
for constructing proofs
based on the probabilistic
method, starting with simple counting and averaging arguments and then introducing
two more advanced tools, the Lovasz local lemma
and the second moment method.
In the context of algorithms
we are generally interested
in explicit constructions
of objects, not merely in proofs'of
existence. In many cases the proofs
of existence
obtained by the
probabiliqtienethod can be converted into efficient
randomized con
struction algorithms. In some
cases, these
proofs can be converted into efficient de
terministic construction algorithms; this process is called derandomization,
since
it
converts a probabilistic argument
into a deterministic one. We give
examples of both
randomized and deterministic construction algorithms arising
from the probabilistic
method.
The Basic Counting
Argument
To prove the existence of an object
with specific properties, we
construct an appropri
ate probability space S of objects and
then show that the probability that
an object in
.S
with the required properties is
selected is strictly greater than 0.
For our first example, we consider the problem of
coloring the edges of
a graph
with
two colors so that there are no large cliques with all
edges having the same colQr. Let
126
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6.I THE
BA,SIC COUNTING ARGUMENT
Knbea
complete graph
1*lttr
utt
(i)
edges) on n vertices. A clique of k vertices in K,,
is a complete subgraPh
K7,.
Theorem
6.1:
If
(b)Zfi>*t
<.
l,
then it is possible to color the edges of Kn with
rwo (l)
colors
so that it has no monochromatic Kp subgraph.
Proof:
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 Fall '09
 Vandam
 Graph Theory, Probability, Probability theory

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