Polylogarithmic Approximation for Minimum Planarization (Almost)

In the minimum planarization problem, given some n-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a log^O(1) n-approximation algorithm for this problem on general graphs with running time n^{O(log n/log log n)}. We also obtain a O(n^ε)-approximation with running time n^O(1/ε) for any arbitrarily small constant ε > 0. Prior to our work, no non-trivial algorithm was known for this problem on general graphs, and the best known result even on graphs of bounded degree was a n^Ω(1)-approximation [1]. As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain O(n^1/2+ε)approximation and n^1/2 log^O(1) n-approximation algorithms in time n^O(1/ε) and n^{O(log n/log log n)} respectively. The previously best-known result was a polynomial-time n^9/10 log^O(1) n-approximation algorithm [2]. Our algorithm introduces several new tools including an efficient grid-minor construction for apex graphs, and a new method for computing irrelevant vertices. Analogues of these tools were previously available only for exact algorithms. Our work gives efficient implementations of these ideas in the setting of approximation algorithms, which could be of independent interest.