The probabilistic method is a
nonconstructive
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existen ...
method, primarily used in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
and pioneered by
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the
probability that the result is of the prescribed kind is strictly greater than zero. Although the proof uses probability, the final conclusion is determined for ''certain'', without any possible error.
This method has now been applied to other areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
such as
number theory,
linear algebra, and
real analysis, as well as in
computer science (e.g.
randomized rounding), and
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
.
Introduction
If every object in a collection of objects fails to have a certain property, then the probability that a random object chosen from the collection has that property is zero.
Similarly, showing that the probability is (strictly) less than 1 can be used to prove the existence of an object that does ''not'' satisfy the prescribed properties.
Another way to use the probabilistic method is by calculating the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of some
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. If it can be shown that the random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value.
Alternatively, the probabilistic method can also be used to guarantee the existence of a desired element in a sample space with a value that is greater than or equal to the calculated expected value, since the non-existence of such element would imply every element in the sample space is less than the expected value, a contradiction.
Common tools used in the probabilistic method include
Markov's inequality, the
Chernoff bound, and the
Lovász local lemma In probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive (possibly small) probability that none of the events will occur. The Lovász local lemma allows on ...
.
Two examples due to Erdős
Although others before him proved theorems via the probabilistic method (for example, Szele's 1943 result that there exist
tournaments containing a large number of
Hamiltonian cycles), many of the most well known proofs using this method are due to Erdős. The first example below describes one such result from 1947 that gives a proof of a lower bound for the
Ramsey number
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say ...
.
First example
Suppose we have a
complete graph on vertices. We wish to show (for small enough values of ) that it is possible to color the edges of the graph in two colors (say red and blue) so that there is no complete subgraph on vertices which is monochromatic (every edge colored the same color).
To do so, we color the graph randomly. Color each edge independently with probability of being red and of being blue. We calculate the expected number of monochromatic subgraphs on vertices as follows:
For any set
of
vertices from our graph, define the variable
to be if every edge amongst the
vertices is the same color, and otherwise. Note that the number of monochromatic
-subgraphs is the sum of
over all possible subsets
. For any individual set
, the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of
is simply the probability that all of the
edges in
are the same color:
:
(S_r^i)
S, or s, is the nineteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ess'' (pronounced ), plural ''esses''.
Histo ...
= 2 \cdot 2^
(the factor of comes because there are two possible colors).
This holds true for any of the
possible subsets we could have chosen, i.e.
ranges from to
. So we have that the sum of
(S_r^i)
S, or s, is the nineteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ess'' (pronounced ), plural ''esses''.
Histo ...
/math> over all
is
:
(S_r^i)
S, or s, is the nineteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ess'' (pronounced ), plural ''esses''.
Histo ...
= 2^.
The sum of expectations is the expectation of the sum (''regardless'' of whether the variables are
independent), so the expectation of the sum (the expected number of all monochromatic
-subgraphs) is
:
(S_r)
S, or s, is the nineteenth Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphab ...
= 2^.
Consider what happens if this value is less than . Since the expected number of monochromatic -subgraphs is strictly less than , there exists a coloring satisfying the condition that the number of monochromatic -subgraphs is strictly less than . The number of monochromatic -subgraphs in this random coloring is a non-negative integer, hence it must be ( is the only non-negative integer less than ). It follows that if
:
(S_r)
S, or s, is the nineteenth Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphab ...
= 2^ < 1
(which holds, for example, for =5 and =4), there must exist a coloring in which there are no monochromatic -subgraphs.
By definition of the
Ramsey number
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say ...
, this implies that must be bigger than . In particular, must
grow at least exponentially with .
A weakness of this argument is that it is entirely
nonconstructive
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existen ...
. Even though it proves (for example) that almost every coloring of the complete graph on vertices contains no monochromatic -subgraph, it gives no explicit example of such a coloring. The problem of finding such a coloring has been open for more than 50 years.
----
Second example
A 1959 paper of Erdős (see reference cited below) addressed the following problem in
graph theory: given positive integers and , does there exist a graph containing only
cycles of length at least , such that the
chromatic number of is at least ?
It can be shown that such a graph exists for any and , and the proof is reasonably simple. Let be very large and consider a random graph on vertices, where every edge in exists with probability . We show that with positive probability, satisfies the following two properties:
:Property 1. contains at most cycles of length less than .
Proof. Let be the number cycles of length less than . Number of cycles of length in the complete graph on vertices is
:
and each of them is present in with probability . Hence by
Markov's inequality we have
:
: Thus for sufficiently large , property 1 holds with a probability of more than .
:Property 2. contains no
independent set of size
.
Proof. Let be the size of the largest independent set in . Clearly, we have
:
when
:
Thus, for sufficiently large , property 2 holds with a probability of more than .
For sufficiently large , the probability that a graph from the distribution has both properties is positive, as the events for these properties cannot be disjoint (if they were, their probabilities would sum up to more than 1).
Here comes the trick: since has these two properties, we can remove at most vertices from to obtain a new graph on
vertices that contains only cycles of length at least . We can see that this new graph has no independent set of size
. can only be partitioned into at least independent sets, and, hence, has chromatic number at least .
This result gives a hint as to why the computation of the
chromatic number of a graph is so difficult: even when there are no local reasons (such as small cycles) for a graph to require many colors the chromatic number can still be arbitrarily large.
See also
*
Interactive proof system
*
Las Vegas algorithm
*
Method of conditional probabilities
*
Probabilistic proofs of non-probabilistic theorems
*
Random graph
Additional resources
Probabilistic Methods in Combinatorics MIT OpenCourseWare
References
* Alon, Noga; Spencer, Joel H. (2000). ''The probabilistic method'' (2ed). New York: Wiley-Interscience. .
*
*
*
J. Matoušek, J. Vondrak
The Probabilistic Method Lecture notes.
* Alon, N and Krivelevich, M (2006)
Extremal and Probabilistic Combinatorics* Elishakoff I., Probabilistic Methods in the Theory of Structures: Random Strength of Materials, Random Vibration, and Buckling, World Scientific, Singapore, , 2017
* Elishakoff I., Lin Y.K. and Zhu L.P. , Probabilistic and Convex Modeling of Acoustically Excited Structures, Elsevier Science Publishers, Amsterdam, 1994, VIII + pp. 296; {{ISBN, 0 444 81624 0
Footnotes
Combinatorics
Mathematical proofs
Probabilistic arguments