LOSSY PLANARIZATION: A CONSTANT-FACTOR APPROXIMATE KERNELIZATION FOR PLANAR VERTEX DELETION

In the \scrF-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 \scrF. This captures numerous important problems including Vertex cover, Feedback vertex set, Treewidth-\eta 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 \scrF-minor-free deletion for the family \scrF = \{K5, K3,3\}. Whenever the family \scrF contains at least one planar graph, then \scrF-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [F. Fomin et al., Proceedings of the 53rd Annual Symposium on Foundations of Computer Science, IEEE, 2012, pp. 470-479]. A polynomial kernelization is a polynomial-time algorithm that, given a graph G and integer k, outputs a graph G^\prime on poly(k) vertices and integer k^\prime, so that \sansO\sansP\sansT(G) \leq k if and only if \sansO\sansP\sansT(G^\prime) \leq k^\prime. The Vertex planarization problem is arguably the simplest setting for which \scrF 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 \alpha-approximate kernelization, for some constant \alpha > 1, based on the framework of lossy kernelization [D. Łokshtanov et al., Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2017, pp. 224-237]. Simply speaking, when given a graph G and integer k, we show how to compute a graph G^\prime on poly(k) vertices so that any \beta-approximate solution to G^\prime can be lifted to an (\alpha \cdot \beta)-approximate solution to G, as long as \sansO\sansP\sansT(G) \leq _\alpha^k_\cdot\beta . In order to achieve this, we develop a toolkit 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-the-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time \scrO(n^\varepsilon)-approximation algorithm, for any \varepsilon > 0, and a quasi-polynomial-time (log n)^\scrO(1)-approximation algorithm, where n is the input size, both randomized [K. Kawarabayashi and A. Sidiropoulos, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2019, pp. 164-175]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively \scrO(\sansO\sansP\sansT^\varepsilon) and (log \sansO\sansP\sansT)^\scrO(1).