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Well-Quasi-Orderings and the Robertson–Seymour Theorems

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Fundamentals of Parameterized Complexity

Part of the book series: Texts in Computer Science ((TCS))

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Abstract

As we will see, well-quasi-orderings (WQOs) provide a powerful engine for demonstrating that classes of problems are FPT. In the next section, we will look at the rudiments of the theory of WQOs, and in subsequent sections, we will examine applications to combinatorial problems.

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Notes

  1. 1.

    Recall that a partial ordering is a quasi-ordering that is also antisymmetric.

  2. 2.

    Sometimes, filters are called upper ideals.

  3. 3.

    Ramsey’s Theorem (Ramsey [572]) states that if B is an infinite set and k is a positive integer, then if we color the subsets of B of size k with colors chosen from {1,…,m}, then there is an infinite subset B′⊆B such that all the size k subsets of B′ have the same color. B′ is referred to a homogeneous subset.

  4. 4.

    Apparently, this result was independently discovered by Pontryagin. We refer the reader to Kennedy, Quintas, and Syslo [437].

  5. 5.

    Chain Minor Ordering was recently shown to be FPT by Blasiok and Kaminski [68].

  6. 6.

    Rado actually proved that this example is canonical in the sense that if Q is a WQO such that Q ω is not a WQO, then the quasi-ordering 〈S,≤〉 of this hint will embed into Q.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Operations Research, Victoria University, Wellington, New Zealand

    Rodney G. Downey

  2. School of Engineering and Information Technology, Charles Darwin University, Darwin, Northern Territory, Australia

    Michael R. Fellows

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  1. Rodney G. Downey
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Downey, R.G., Fellows, M.R. (2013). Well-Quasi-Orderings and the Robertson–Seymour Theorems. In: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-5559-1_17

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  • DOI: https://doi.org/10.1007/978-1-4471-5559-1_17

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