Polylogarithmic approximation for minimum planarization (almost)

TL;DR

This paper introduces polylogarithmic approximation algorithms for the minimum planarization problem, significantly improving previous results and also enhancing approximation algorithms for the crossing number problem on bounded degree graphs.

Contribution

It presents the first non-trivial approximation algorithms for general graphs in minimum planarization, using new tools like grid-minor construction and irrelevant vertices computation.

Findings

Polylogarithmic approximation for minimum planarization.

Improved approximation algorithms for crossing number problem.

Introduction of new tools for approximation algorithms in topological graph theory.

Abstract

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 $O(n^\varepsilon)$-approximation with running time $n^{O(1/\varepsilon)}$ for any arbitrarily small constant $\varepsilon > 0$. 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 $n^{\Omega(1)}$-approximation [Chekuri and Sidiropoulos 2013]. As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain $O(n^{1/2+\varepsilon})$-approximation and $n^{1/2} \log^{O(1)} n$-approximation algorithms in time $n^{O(1/\varepsilon)}$ 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 [Chuzhoy 2011]. 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.