The Probabilistic Method

Transcription

1 The Probabilistic Method In Graph Theory Ehssan Khanmohammadi Department of Mathematics The Pennsylvania State University February 25, 2010

2 What do we mean by the probabilistic method? Why use this method? The Probabilistic Method and Paul Erdős The probabilistic method is a technique for proving the existence of an object with certain properties by showing that a random object chosen from an appropriate probability distribution has the desired properties with positive probability. Pioneered and championed by Paul Erdős who applied it mainly to problems in combinatorics and number theory from 1947 onwards.

3 What do we mean by the probabilistic method? Why use this method? An apocryphal story quoted from Molloy and Reed At every combinatorics conference attended by Erdős in 1960s and 1970s, there was at least one talk which concluded with Erdős informing the speaker that almost every graph was a counterexample to his/her conjecture!

4 What do we mean by the probabilistic method? Why use this method? Three facts about the probabilistic method which are worth bearing in mind: 1: Large and Unstructured Output Graphs The probabilistic method allows us to consider graphs which are both large and unstructured. The examples constructed using the probabilistic method routinely contain many, say 10 10, nodes. Explicit constructions necessarily introduce some structuredness to the class of graphs built, which thus restricts the graphs considered.

5 What do we mean by the probabilistic method? Why use this method? 2: Powerful and Easy to Use Erdős would routinely perform the necessary calculations to disprove a conjecture in his head during a fifteen-minute talk.

6 What do we mean by the probabilistic method? Why use this method? 2: Powerful and Easy to Use Erdős would routinely perform the necessary calculations to disprove a conjecture in his head during a fifteen-minute talk. 3: Covers almost Every Graph Erdős did not say some graph is a counterexample to your conjecture, but rather almost every graph is a counterexample to your conjecture.

7 Applications in Discrete Mathematics One can classify the applications of probabilistic techniques in discrete mathematics into two groups. 1: Study of Random Objects (Graphs, Matrices, etc.) A typical problem is the following: if we pick a graph at random, what is the probability that it contains a Hamiltonian cycle?

8 2: Proof of the Existence of Certain Structures Choose a structure randomly (from a probability distribution that you are free to specify). Estimate the probability that it has the properties you want. Show that this probability is greater than 0, and therefore conclude that such a structure exists.

9 2: Proof of the Existence of Certain Structures Choose a structure randomly (from a probability distribution that you are free to specify). Estimate the probability that it has the properties you want. Show that this probability is greater than 0, and therefore conclude that such a structure exists. Surprisingly often it is much easier to prove this than it is to give an example of a structure that works.

10 Example 1: Szele s result Two Definitions A directed complete graph is called a tournament. By a Hamiltonian path in a tournament we mean a path which traces each node (exactly once) following the direction of the graph.

11 Example 1: Szele s result Two Definitions A directed complete graph is called a tournament. By a Hamiltonian path in a tournament we mean a path which traces each node (exactly once) following the direction of the graph. The following result of Szele (1943) is ofttimes considered the first use of the probabilistic method. Theorem (Szele 1943) There is a tournament T with n players and at least n!2 (n 1) Hamiltonian paths.

12 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes.

13 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths.

14 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n.

15 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path.

16 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path. P(X σ = 1) = 2 (n 1).

17 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path. P(X σ = 1) = 2 (n 1). X = σ X σ, thus E(X ) = σ E(X σ) = n!2 (n 1).

18 Szele s result (cont.) Proof. Choose uniform distribution on all tournaments with n nodes. Let X be the random variable counting the number of Hamiltonian paths. A permutation σ on the set of nodes defines a Hamiltonian path iff (σ(i), σ(i + 1)) is a directed edge for all 1 i < n. Let X σ be the indicator of σ defines a Hamiltonian path. P(X σ = 1) = 2 (n 1). X = σ X σ, thus E(X ) = σ E(X σ) = n!2 (n 1). Conclusion: There is a tournament for which X is equal to at least E(X ).

19 Remark 1 A player who wins all games would naturally be the tournament s winner. However, there might not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament T = (V, E) is called k-paradoxical if for every k-element subset S of V there is a vertex v 0 in V \ S such that v 0 v for each v S. By means of the probabilistic method Erdős showed that, for any fixed value of k, if V is sufficiently large, then almost every tournament on V is k-paradoxical.

20 Remark 1 A player who wins all games would naturally be the tournament s winner. However, there might not be such a player. A tournament for which every player loses at least one game is called a 1-paradoxical tournament. More generally, a tournament T = (V, E) is called k-paradoxical if for every k-element subset S of V there is a vertex v 0 in V \ S such that v 0 v for each v S. By means of the probabilistic method Erdős showed that, for any fixed value of k, if V is sufficiently large, then almost every tournament on V is k-paradoxical. Remark 2 Szele conjectured that the maximum possible number of Hamiltonian paths in a tournament on n players is at most n! (2 o(1)) n. Alon proved this conjecture in 1990 using the probabilistic method.

21 Example 2: Lower bound for diagonal Ramsey numbers R(k, k) Definition The Ramsey number R(k, l) is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k or there is a blue K l.

22 Example 2: Lower bound for diagonal Ramsey numbers R(k, k) Definition The Ramsey number R(k, l) is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k or there is a blue K l. Ramsey (1929) showed that R(k, l) is finite for any two integers k, l.

23 Example 2: Lower bound for diagonal Ramsey numbers R(k, k) Definition The Ramsey number R(k, l) is the smallest integer n such that in any two-coloring of the edges of a complete graph on n vertices K n by red and blue, either there is a red K k or there is a blue K l. Ramsey (1929) showed that R(k, l) is finite for any two integers k, l. Theorem (Erdős (1947)) If ( n k) 2 1 ( k 2) < 1, then R(k, k) > n. Thus R(k, k) > 2 k/2 for each k 3.

24 Lower bound for R(k, k) (cont.) Proof. Consider a random 2-coloring of K n : Color each edge independently with probability 1 2 of being red and 1 2 of being blue.

25 Lower bound for R(k, k) (cont.) Proof. Consider a random 2-coloring of K n : Color each edge independently with probability 1 2 of being red and 1 2 of being blue. For any fixed set R of k nodes, let X R be the indicator of being monochromatic for induced subgraph of R, and define X for the whole graph similarly.

26 Lower bound for R(k, k) (cont.) Proof. Consider a random 2-coloring of K n : Color each edge independently with probability 1 2 of being red and 1 2 of being blue. For any fixed set R of k nodes, let X R be the indicator of being monochromatic for induced subgraph of R, and define X for the whole graph similarly. Clearly, P(X R = 1) = 2 1 (k 2), and by our assumption E(X ) = ( ) n E(X R ) = 2 1 (k 2) < 1 k R

27 Proof continued. E(X ) < 1, thus, there exists a complete graph on n nodes with no monochromatic subgraph on k nodes, because the expected value, that is, the mean number of monochromatic subgraphs is less than one, where the mean is taken over all 2-colorings of K n. So, R(k, k) > n.

28 Proof continued. E(X ) < 1, thus, there exists a complete graph on n nodes with no monochromatic subgraph on k nodes, because the expected value, that is, the mean number of monochromatic subgraphs is less than one, where the mean is taken over all 2-colorings of K n. So, R(k, k) > n. Note that if k 3 and we take n = 2 k/2, then ( ) n (k 2) 1+ k 2 < k k! and hence R(k, k) > 2 k/2. n k 2 k2 /2 < 1,

29 Example 3: A Result of Caro and Wei A Definition and a Notation A subset of the nodes of a graph is called independent if no two of its elements are adjacent. The size of a maximal (with respect to inclusion) independent set in a graph G = (V, E) is denoted by α(g).

30 Example 3: A Result of Caro and Wei A Definition and a Notation A subset of the nodes of a graph is called independent if no two of its elements are adjacent. The size of a maximal (with respect to inclusion) independent set in a graph G = (V, E) is denoted by α(g). Theorem (Caro (1979), Wei (1981)) α(g) v V 1 d v +1.

31 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }.

32 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }. Let X v be the indicator random variable for v I and X = v V X v = I.

33 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }. Let X v be the indicator random variable for v I and X = v V X v = I. For each v, E(X v ) = P(v I ) = 1 d v +1, since v I iff v is the least element among v and its neighbors.

34 Proof. Let < be a uniformly chosen total ordering of V. Define I = { v V { v, w } E v < w }. Let X v be the indicator random variable for v I and X = v V X v = I. For each v, E(X v ) = P(v I ) = 1 d v +1, since v I iff v is the least element among v and its neighbors. 1 d v +1 Hence E(X ) = v V, and so there exists a specific ordering < with I v V 1 d v +1.

35 Explicit Constructions and Algorithmic Aspects The problem of finding a good explicit construction is often very difficult. Even the simple proof of Erdős that there are red/blue colorings of graphs with 2 k/2 nodes containing no monochromatic clique of size k leads to an open problem that seems very difficult. An Open Problem Can we explicitly construct a graph as described above with n (1 + ɛ) k nodes in time that is polynomial in n? This problem is still wide open, despite considerable efforts from many mathematicians.

36 Thank You! References: 1 Alon, Spencer, The Probabilistic Method. 2 Gowers, et al., The Princeton Companion to Mathematics. 3 Molloy, Reed, Graph Colouring and the Probabilistic Method.