  Parameterized complexity allows us to analyze the time complexity of problems
with respect to a natural parameter depending on the problem. Reoptimization
looks for solutions or approximations for problem instances when given
solutions to neighboring instances. We try to combine both techniques, in order
to better classify the complexity of problems in the parameterized setting.
Specifically, we see that some problems in the class of compositional problems,
which do not have polynomial kernels under standard complexity-theoretic
assumptions, do have polynomial kernels under reoptimization for some local
modifications. Moreover, we find that the reoptimization version of Vertex
Cover has a polynomial kernel of size 2k using crown decomposition. Finally, in
a negative result, we prove that the reoptimization version of Connected Vertex
Cover does not have a Turing kernelization unless Set Cover has a polynomial
kernel
