  We give an algebraic, determinant-based algorithm for the K-Cycle problem,
i.e., the problem of finding a cycle through a set of specified elements. Our
approach gives a simple FPT algorithm for the problem, matching the
$O^*(2^{|K|})$ running time of the algorithm of Bj\"orklund et al. (SODA,
2012). Furthermore, our approach is open for treatment by classical algebraic
tools (e.g., Gaussian elimination), and we show that it leads to a polynomial
compression of the problem, i.e., a polynomial-time reduction of the $K$-Cycle
problem into an algebraic problem with coding size $O(|K|^3)$. This is
surprising, as several related problems (e.g., k-Cycle and the Disjoint Paths
problem) are known not to admit such a reduction unless the polynomial
hierarchy collapses. Furthermore, despite the result, we are not aware of any
witness for the K-Cycle problem of size polynomial in $|K|+\log n$, which seems
(for now) to separate the notions of polynomial compression and polynomial
kernelization (as a polynomial kernelization for a problem in NP necessarily
implies a small witness).
