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Recent advances in well-partial-order theory, especially the seminal contributions of Robertson and Seymour, have troubling consequences for those who would equate tractability with polynomial-time decidability. Specifically:many problems are now known to be decidable in low-degree polynomial time, but only by decision algorithms with overwhelmingly astronomical constants of proportionality,the existence of such a polynomial-time decision algorithm alone does not ensure that a corresponding search problem can be solved efficiently, andeven if both a decision problem and a corresponding search problem can be shown to be polynomial-time computable, there is no guarantee that correct algorithms can be found or even recognized within any bounded amount of time.In this paper, we present a number of techniques for dealing with these remarkable features of algorithms based on well-partially-ordered sets. Our main results include a general strategy with which such algorithms can be made constructive. With the aid of this method, we demonstrate that low-degree polynomial-time algorithms are now known for almost all of the catalogued applications of RS posets. We also prove that, despite the nonconstructive nature of the well-partial-order theory on which this line of research is based, no RS poset application can settle P @@@@ N P non-constructively by any established method of argument.

On search decision and the efficiency of polynomial-time algorithms