Let F be a finite family of graphs. In the F-Deletion problem, one is given a graph G and an integer k, and the goal is to find k vertices whose deletion results in a graph with no minor from the family F. This may be regarded as a far-reaching generalization of Vertex Cover and Feedback vertex Set. In their seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains a planar graph. As the size of their kernel is g(F) * k^{f(F)}, a natural follow-up question was whether the dependence on F in the exponent of k can be avoided. The answer turned out to be negative: Giannapoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] proved that this is already inevitable for the special case of the Treewidth-d-Deletion problem. In this work, we show that this non-uniformity can be avoided at the expense of a small loss. First, we present a simple 2-approximate kernelization algorithm for Treewidth-d-Deletion with kernel size g(d) * k^5. Next, we show that the approximation factor can be made arbitrarily close to 1, if we settle for a kernelization protocol with O(1) calls to an oracle that solves instances of size bounded by a uniform polynomial in k. We also obtain linear kernels on sparse graph classes when F contains a planar graph, whereas the previously known theorems required all graphs in F to be connected. Specifically, we generalize the kernelization algorithm by Kim, Langer, Paul, Reidl, Rossmanith, Sau & Sikdar [TALG 2015] on graph classes that exclude a topological minor.

翻译：令 F 为一个有限图族。在 F-删除问题中，给定一个图 G 和一个整数 k，目标是找到 k 个顶点，删除这些顶点后得到的图不包含来自族 F 的任何子式。这可以被视为顶点覆盖和反馈顶点集问题的一个深远推广。在其开创性工作中，Fomin、Lokshtanov、Misra 和 Saurabh [FOCS 2012] 给出了当族 F 包含一个平面图时该问题的多项式核。由于其核的大小为 g(F) * k^{f(F)}，一个自然的后续问题是：k 的指数中对 F 的依赖是否可以避免。答案被证明是否定的：Giannapoulou、Jansen、Lokshtanov 和 Saurabh [TALG 2017] 证明了对于树宽-d-删除这一特例，这种非均匀性已是不可避免的。在本工作中，我们表明可以通过承受微小的损失来避免这种非均匀性。首先，我们为树宽-d-删除问题提出了一个简单的 2-近似核化算法，其核大小为 g(d) * k^5。其次，我们证明，如果我们接受一个核化协议，该协议需要 O(1) 次调用一个预言机来解决规模受限于 k 的一致多项式的实例，那么近似因子可以任意接近 1。此外，当 F 包含一个平面图时，我们在稀疏图类上获得了线性核，而先前已知的定理要求 F 中的所有图都是连通的。具体而言，我们推广了 Kim、Langer、Paul、Reidl、Rossmanith、Sau 和 Sikdar [TALG 2015] 在排除拓扑子式的图类上的核化算法。