Designing Practical PTASes for Minimum Feedback Vertex Set in Planar Graphs

TL;DR

This paper introduces two algorithms for the minimum feedback vertex set problem in planar graphs: a PTAS with high accuracy but longer runtime, and a heuristic that balances solution quality and efficiency, outperforming existing methods in many cases.

Contribution

The paper presents a novel PTAS and a heuristic algorithm for planar graphs, improving solution quality and providing practical trade-offs compared to previous approaches.

Findings

The PTAS is competitive with the 2-approximation in large graphs.

The heuristic often outperforms the 2-approximation.

The heuristic offers a trade-off between runtime and solution quality.

Abstract

We present two algorithms for the minimum feedback vertex set problem in planar graphs: an $O(n \log n)$ PTAS using a linear kernel and balanced separator, and a heuristic algorithm using kernelization and local search. We implemented these algorithms and compared their performance with Becker and Geiger's 2-approximation algorithm. We observe that while our PTAS is competitive with the 2-approximation algorithm on large planar graphs, its running time is much longer. And our heuristic algorithm can outperform the 2-approximation algorithm on most large planar graphs and provide a trade-off between running time and solution quality, i.e. a "PTAS behavior".