Lossy planarization: a constant-factor approximate kernelization for planar vertex deletion

In the F-minor-free deletion problem we are given an undirected graph G and the goal is to find a minimum vertex set that intersects all minor models of graphs from the family F. This captures numerous important problems including Vertex cover, Feedback vertex set, Treewidth-• modulator, and Vertex planarization. In the latter one, we ask for a minimum vertex set whose removal makes the graph planar. This is a special case of F-minor-free deletion for the family F = {K5, K3,3}. Whenever the family F contains at least one planar graph, then F-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [Fomin, Lokshtanov, Misra, and Saurabh, FOCS'12]. A polynomial kernelization is a polynomial-time algorithm that, given a graph G and integer k, outputs a graph G′ on poly(k) vertices and integer k′, so that OPT(G) ≤ k if and only if OPT(G′) ≤ k′. The Vertex planarization problem is arguably the simplest setting for which F does not contain a planar graph and the existence of a constant-factor approximation or a polynomial kernelization remains a major open problem. In this work we show that Vertex planarization admits an algorithm which is a combination of both approaches. Namely, we present a polynomial α-approximate kernelization, for some constant α > 1, based on the framework of lossy kernelization [Lokshtanov, Panolan, Ramanujan, and Saurabh, STOC'17]. Simply speaking, when given a graph G and integer k, we show how to compute a graph G′ on poly(k) vertices so that any β-approximate solution to G′ can be lifted to an (α· β)-approximate solution to G, as long as OPT(G) ≤ k/α· β. In order to achieve this, we develop a framework for sparsification of planar graphs which approximately preserves all separators and near-separators between subsets of the given terminal set. Our result yields an improvement over the state-of-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time O(n")-approximation algorithm, for any "> 0, and a quasi-polynomial-time (logn)O(1)-approximation algorithm, where n is the input size, both randomized [Kawarabayashi and Sidiropoulos, FOCS'17]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively O(OPT") and (logOPT)O(1).