A tight lower bound for Vertex Planarization on graphs of bounded treewidth

TL;DR

This paper establishes a tight lower bound for the Vertex Planarization problem on graphs with bounded treewidth, showing that current algorithms are essentially optimal under the Exponential Time Hypothesis.

Contribution

It proves that the existing algorithm's running time is tight, indicating that significant improvements for single-exponential algorithms are unlikely.

Findings

The current algorithm's time complexity is proven to be optimal under ETH.

Improving the algorithm to a single-exponential dependency on solution size is unlikely.

The lower bound holds even for unweighted graphs and when parameterized by treedepth.

Abstract

In the Vertex Planarization problem one asks to delete the minimum possible number of vertices from an input graph to obtain a planar graph. The parameterized complexity of this problem, parameterized by the solution size (the number of deleted vertices) has recently attracted significant attention. The state-of-the-art algorithm of Jansen, Lokshtanov, and Saurabh [SODA 2014] runs in time $2^{O(k \log k)} \cdot n$ on $n$-vertex graph with a solution of size $k$. It remains open if one can obtain a single-exponential dependency on $k$ in the running time bound. One of the core technical contributions of the work of Jansen, Lokshtanov, and Saurabh is an algorithm that solves a weighted variant of Vertex Planarization in time $2^{O(w \log w)} \cdot n$ on graphs of treewidth $w$. In this short note we prove that the running time of this routine is tight under the Exponential Time Hypothesis, even in unweighted graphs and when parameterizing by treedepth. Consequently, it is unlikely that a potential single-exponential algorithm for Vertex Planarization parameterized by the solution size can be obtained by merely improving upon the aforementioned bounded treewidth subroutine.