Ramsey Theory and Applications

CS 594 Graph TheoryPresented by: Kai Wang

Backgrounds and Definitions

From Pigeonhole Principle to Ramsey Theory

If more than pigeons are put into pigeonholes, then there is a pigeonhole that contains at least two pigeons.

If objects are partitioned in classes, then some class contains at least objects and some class contains at most objects.

If objects are partitioned into classes with quotas , then some class meets its quota.

Ramsey Theory is a generalization of pigeonhole principle.

What is this all about?

Part of extremal combinatorics: smallest configuration with special structural properties.

In extremal graph theory, many problems are difficult.

Applications of Pigeonhole Principle Pumping lemma for regular

languages and context free languages.

Every sequence of distinct numbers contains a monotone sub-sequence of length .

Every (n+1)-subset of [2n]={1,2,…,2n} contains two elements that are coprime and contains two elements such that one divides the other.

History

Historical Perspectives

Ramsey theory was initially studied in the context of propositional logic (1928)

Theodore S. Motzkin: “Complete disorder is impossible”.

Become known after Paul Erdos and George Szekeres (1935) applied it in graph theory

Some Notations

[N]={1,2,…,N} denotes the set of -subsets of the

set . For example, ={{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Ramsey Theorem

Given positive integers and , there exists an integer such that for every -coloring of , there is a subset of [] of size whose -subsets all receive the same color for some .

The smallest such integer is called Ramsey number .

Interesting proof technique: double induction!

Ramsey Number Examples

R(3,3;2)=6 R(3,3,3;2)=17 R(4,4;3)=13 We don’t know the exact value of

R(5,5;2), although it is between 43 and 49.

Little is known about how to calculate the exact value of Ramsey numbers, because currently we only know of the brute-force method, which is impractical.

Graph Ramsey Theory

Given simple graphs , the graph Ramsey number is the smallest integer such that every -coloring of (the edge set of ) contains a copy of in color for some .

Ramsey theorem ensures that graph Ramsey number must exist.

Graph Ramsey Number Examples because in case of complete

graphs, this is equivalent. , where is a tree with vertices.

Applications

Potential in areas that can apply pigeon hole principles

Graph Ramsey theory adds many natural examples to computational complexity classes

A decision problem

Given graphs , does every edge-coloring of with red and blue contains either a red or a blue ?

coNP for fixed and . -complete in general.

Another decision problem

Given graphs , does every edge-coloring of with red and blue contains either a red or a blue as an induced subgraph?

-complete in general.

Application in information retrieval problem Given a set of distinct keys from

key space build a data structure to store so that membership queries of the form “Is in ?” can be answered quickly.

If is sufficiently large, then queries are needed in the worst case with any table structure.

Part II

Edmonds’ maximum matching algorithm

Maximum Matching Problem A problem once thought to be

difficult, but finally found to be in in 1960s by Edmonds.

Edmonds first drew the “easy-hard” boundary between problems.

Berge’s Lemma

An augmenting path in a graph G with respect to a matching M is an alternating path with the two endpoints exposed (unmatched).

A matching M in G is of maximum cardinality if and only if (G,M) does not contain an augmenting path.

Key Concepts: Blossom

(courtesy of http://en.wikipedia.org/wiki/File:Edmonds_blossom.svg)

Blossom Algorithm

If the contracted graph is G’, then a maximum matching of G’ corresponds to a maximum matching of G.

We first get a maximum matching M’ of G’, then by expanding M’ we get a maximum matching M of G.

Complexity: O(n^4)

Other Implementations

One of the famous is Gabow’s labeling implementation of Edmonds’ algorithm by avoiding expanding M’ in the contracted graph G’.

Complexity: O(n^3)

References

Douglas B. West: Introduction to Graph Theory, Section 8.3

Vera Rosta: Ramsey Theory Applications, The Electronic Journal of Combinatorics, 2004

Edmonds, Jack (1965). "Paths, trees, and flowers". Canad. J. Math. 17: 449–467

Harold N. Gabow: An Efficient implementation of Edmonds' Algorithm for Maximum Matching on Graphs, Journal of the ACM, Volume 23 Issue 2, 1976

Homework

Prove or disprove: R(p,2;2)=p when p>=2.

Prove or disprove: R(4,3;2)>=10. What is the relationship between

maximum matching size and minimum vertex cover size?

Thank You

Questions?