An FPT Variant Of The Shadow Problem With Kernelization

The shadow problem (SIS) gets as input a forest F , and a map that assigns subtrees, called shadows, to leaves of F . SIS asks whether there exists a set of |F | leaves, one from each tree, such that no leaf lies in the shadow of another. Usually SIS is considered as a parameterized problem with parameter k bounding the cardinality of F , for which some fixedparameter tractability time bounds have been proven, namely O(nk) in [2] and O(n3) in [4], where n is the number of vertices in F . In this paper, we discuss a variant of SIS that essentially is characterized through a different parameterization using two independent parameters, namely k as above, and s bounding the shadow size. We provide a kernelization w.r.t. this parameterization, and prove a fixed-parameter tractability bound of O(k · n + p(k, s)3) where p is a polynomial in the parameters k, s.