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Local Optimization Algorithms for Maximum Planar Subgraph

Authors Gruia Călinescu , Sumedha Uniyal

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • planar graph
  • maximum subgraph
  • approximation algorithm
  • matroid parity
  • local optimization

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Abstract

Consider the NP-hard problem of, given a simple graph G, to find a planar subgraph of G with the maximum number of edges. This is called the Maximum Planar Subgraph problem and the best known approximation is 4/9 and is obtained by sophisticated Graphic Matroid Parity algorithms. Here we show that applying a local optimization phase to the output of this known algorithm improves this approximation ratio by a small {ε} = 1/747 > 0. This is the first improvement in approximation ratio in more than a quarter century. The analysis relies on a more refined extremal bound on the Lovász cactus number in planar graphs, compared to the earlier (tight) bound of [Gruia Călinescu et al., 1998; Chalermsook et al., 2019].
A second local optimization algorithm achieves a tight ratio of 5/12 for Maximum Planar Subgraph without using Graphic Matroid Parity. We also show that applying a greedy algorithm before this second optimization algorithm improves its ratio to at least 91/216 < 4/9. The motivation for not using Graphic Matroid Parity is that it requires sophisticated algorithms that are not considered practical by previous work. The best previously published [Chalermsook and Schmid, 2017] approximation ratio without Graphic Matroid Parity is 13/33 < 5/12.

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Gruia Călinescu and Sumedha Uniyal. Local Optimization Algorithms for Maximum Planar Subgraph. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.38

Author Details

Gruia Călinescu
  • Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA
Sumedha Uniyal
  • Department of Computer Science, Aalto University, Finland

Funding

Gruia Călinescu’s work performed in part while visiting Northwestern University. Sumedha’s work performed in part at Aalto University, Finland, and in part at Northwestern University, United States of America. She was supported by the Academy of Finland (grant agreement number 314284).
  • Uniyal, Sumedha: Sumedha Uniyal passed away in February 2020. Her husband, Prajna Uniyal, email uniyal.prajna@gmail.com, has approved this submission.

Acknowledgements

Gruia thanks Michael Pelsmajer for many comments that improved the presentation. Gruia thanks the anonymous SODA 2024 reviewer B who suggested using a positive linear combination (instead of case analysis) when computing the ε-improvement of Algorithm MTLK4, which lead to a slightly simpler proof with a bigger ε. Gruia also thanks Konstantin Makarychev for hints on finding the best linear combination.

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