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Ramsey Theory and Applications CS 594 Graph Theory Presented by: Kai Wang

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Backgrounds and Definitions

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From Pigeonhole Principle to Ramsey Theory

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What is this all about? Part of extremal combinatorics: smallest configuration with special structural properties. In extremal graph theory, many problems are difficult.

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Applications of Pigeonhole Principle

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History

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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

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Some Notations

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Ramsey Theorem

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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.

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Graph Ramsey Theory

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Graph Ramsey Number Examples

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Applications Potential in areas that can apply pigeon hole principles Graph Ramsey theory adds many natural examples to computational complexity classes

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A decision problem

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Another decision problem

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Application in information retrieval problem

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Part II Edmonds’ maximum matching algorithm

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Maximum Matching Problem

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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.

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

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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)

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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)

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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

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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?

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Thank You Questions?

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