Approximately Interpolating Between Uniformly and Non-Uniformly Polynomial Kernels

The problem of computing a minimum set of vertices intersecting a finite set of forbidden minors in a given graph is a fundamental graph problem in the area of kernelization with numerous well-studied special cases. A major breakthrough in this line of research was made by Fomin et al. [FOCS 2012], who showed that the ?-Treewidth Modulator problem (delete minimum number of vertices to ensure that treewidth is at most ?) has a polynomial kernel of size kg(?) for some function g. A second standout result in this line is that of Giannapoulou et al. [ACM TALG 2017], who obtained an f(?)kO(1)-size kernel (for some function f) for the ?-Treedepth Modulator problem (delete fewest number of vertices to make treedepth at most ?) and showed that some dependence of the exponent of k on ? in the result of Fomin et al. for the ?-Treewidth Modulator problem is unavoidable under reasonable complexity hypotheses. In this work, we provide an approximate interpolation between these two results by giving, for every > 0, a (1+)-approximate kernel of size f'(?, ?, 1/?) kg'(?) (for some functions f' and g') for the problem of deciding whether k vertices can be deleted from a given graph to obtain a graph that has elimination distance at most ? to the class of graphs that have treewidth at most ?. Graphs of treedepth ? are precisely the graphs with elimination distance at most ? 1 to the graphs of treewidth 0 and graphs of treewidth ? are simply graphs with elimination distance 0 to graphs of treewidth ?. Consequently, our result "approximately" interpolates between these two major results in this active line of research.