Lossy Planarization: A Constant-Factor Approximate Kernelization for Planar Vertex Deletion

TL;DR

This paper introduces a novel lossy kernelization approach for the Vertex Planarization problem, achieving better approximation ratios by combining kernelization with existing algorithms.

Contribution

It presents the first polynomial approximate kernelization for Vertex Planarization, leveraging a new framework for sparsification of planar graphs.

Findings

Achieves polynomial A-approximate kernelization for Vertex Planarization.

Improves approximation factors to O(OPT^eps) and (log OPT)^O(1) using pipelined algorithms.

Develops a framework for sparsification that preserves separators in planar graphs.

Abstract

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).