In this paper, we prove a new scaling lemma for vertex weighted minor free graphs that allows for a smooth trade-off between the weight of a vertex set S and the treewidth of G - S. More precisely, we show the following. There exists an algorithm that given an H-minor free graph G, a weight function w : V(G) -> Q(+) and integers t and s, runs in polynomial time, and outputs a subset S subset of V(G) of weight at most d log n . opt (G, w, t)/s such that the treewidth of G - S is at most c.st. Here, d and c are fixed constants that depend only on H, and opt (G, w, t) is the (unknown) minimum weight of a subset U subset of V(G) such that the treewidth of G - U is at most t. This lemma immediately yields the first polynomialtime approximation schemes (PTASes) for WEIGHTED TREEWIDTH-eta VERTEX DELETION, for eta >= 2, on graphs of bounded genus and the first PTAS for WEIGHTED FEEDBACK VERTEX SET on H-minor free graphs. These results effortlessly generalize to include weighted edge deletion problems, to all WEIGHTED CONNECTED PLANAR F-DELETION problems, and finally to quasi polynomial time approximation schemes (QPTASes) for all of these problems on H-minor free graphs. For most of these problems even constant factor approximation algorithms, even on planar graphs, were not previously known. Additionally, using the scaling lemma we subsume, simplify and extend the recent framework of Cohen-Addad et al. [STOC 2016] for turning constant factor approximation algorithms for "ubiquitous" problems into PTASes for the same problems on graphs of bounded genus. Specifically, we obtain PTASes for ubiquitous problems without the requirement of having a constant factor approximation. While the statement of the scaling lemma is inspired by an analogous lemma by Cohen-Addad et al. [STOC 2016] for edge contractions on weighted graphs of bounded genus, as well as a scaling lemma by Fomin et al. [SODA 2011] for unweighted graphs, the proof is entirely different. The proof detours via three different linear programming relaxations for the WEIGHTED TREEWIDTH-eta VERTEX DELETION problems and a strengthening of a recent rounding procedure of Bansal et al. [SODA 2017] enhanced by the classic Klein-Plotkin-Rao Theorem [STOC 1993].
