In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the family F = {K_5, K_{3,3}}. 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]. 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 A-approximate kernelization, for some constant A > 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 B-approximate solution to G' can be lifted to an (A*B)-approximate solution to G, as long as A*B*OPT(G) <= k. 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^eps)-approximation algorithm, for any eps > 0, and a quasi-polynomial-time (log n)^O(1) approximation algorithm, both randomized [Kawarabayashi and Sidiropoulos, FOCS'17]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively O(OPT^eps) and (log OPT)^O(1).
翻译:在F-minor删除问题中, 我们想要在给定的图表中找到一个最小的顶点, 将家庭 F. 的图解图解中的所有微小模型相交。 Vertex 的平流化问题是家庭 F = {K_ 5, K ⁇ 3, ⁇ 。 当家庭 F 包含至少一个平面图时, 那么F- 平面删除就会被人们所知道的常态近效近端算法和多式内分解 [Fomin, Lokshtanov, Misra, Saurabah, FOSC'12] 。 Vertex 平面图化问题可能是一个最简单的设置, F = = {K_ 5, K_ 3 {\\\\\ 。 只要家庭 Fern- 平面图解算中至少有一个常态离子的近似近似点, 那么Vetricalation 将一个算法化, 和多式平面平面体- hal- dal- dal- pal- palizional- pal- palization, lax- us a lax- sal- lax a lax lax a lax lax a lax a lax a lax.