  The gist of many (NP-)hard combinatorial problems is to decide whether a
universe of $n$ elements contains a witness consisting of $k$ elements that
match some prescribed pattern. For some of these problems there are known
advanced algebra-based FPT algorithms which solve the decision problem but do
not return the witness. We investigate techniques for turning such a
YES/NO-decision oracle into an algorithm for extracting a single witness, with
an objective to obtain practical scalability for large values of $n$. By
relying on techniques from combinatorial group testing, we demonstrate that a
witness may be extracted with $O(k\log n)$ queries to either a deterministic or
a randomized set inclusion oracle with one-sided probability of error.
Furthermore, we demonstrate through implementation and experiments that the
algebra-based FPT algorithms are practical, in particular in the setting of the
$k$-path problem. Also discussed are engineering issues such as optimizing
finite field arithmetic.
