Polynomial Time Recognition
The Robertson–Seymour theorem has an important consequence in computational complexity, due to the proof by Robertson and Seymour that, for each fixed graph G, there is a polynomial time algorithm for testing whether larger graphs have G as a minor. The running time of this algorithm can be expressed as a cubic polynomial in the size of the larger graph (although there is a constant factor in this polynomial that depends superpolynomially on the size of G). As a result, for every minor-closed family F, there is polynomial time algorithm for testing whether a graph belongs to F: simply check, for each of the forbidden minors for F, whether the given graph contains that forbidden minor.
However, this method requires a specific finite obstruction set to work, and the theorem does not provide one. The theorem proves that such a finite obstruction set exists, and therefore the problem is polynomial because of the above algorithm. However, the algorithm can be used in practice only if such a finite obstruction set is provided. As a result, the theorem proves that the problem can be solved in polynomial time, but does not provide a concrete polynomial-time algorithm for solving it. Such proofs of polynomiality are non-constructive: they prove polynomiality of problems without providing an explicit polynomial-time algorithm. In many specific cases, checking whether a graph is in a given minor-closed family can be done more efficiently: for example, checking whether a graph is planar can be done in linear time.
Read more about this topic: Robertson–Seymour Theorem
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