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International Workshop on Combinatorial Algorithms

IWOCA 2016: Combinatorial Algorithms pp 334–346Cite as

Limits of Greedy Approximation Algorithms for the Maximum Planar Subgraph Problem

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9843))

Abstract

The Maximum Planar Subgraph (MPS) problem asks for a planar subgraph with maximum edge cardinality of a given undirected graph. It is known to be MaxSNP-hard and the currently best known approximation algorithm achieves a ratio of 4/9.

We analyze the general limits of approximation algorithms for MPS, based either on planarity tests or on greedy inclusion of certain subgraphs. On the one hand, we cover upper bounds for the approximation ratios. On the other hand, we show NP-hardness for thereby arising subproblems, which hence must be approximated themselves. We also provide simpler proofs for two already known facts.

M. Chimani and T. Wiedera—Supported by the German Research Foundation (DFG) project CH 897/2-1.

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Notes

  1. 1.

    The proof by Cimikowski [4] for the Hopcroft-Tarjan based heuristic exploits the specific embedding of backedges and cannot be generalized to arbitrary algorithms based on DFS trees.

  2. 2.

    We start at \(\pi _0\) with \(\pi _0\pi _1\). Next, we pick all edges of \(B_0\) that are incident to \(\pi _1\) since the \(\mathcal {I}(B_0)\)-vertices lead only to \(\pi _0\) (visited). We iterate this until we arrive at \(\pi _{n-2}\pi _{n-1}\). Finally, we pick all edges connecting \(\pi _{n-1}\) with \(\mathcal {I}(B_{n-2})\) and \(\mathcal {I}(B_{n-1})\).

  3. 3.

    Starting at r (level 0) includes all edges of \(E_r\). E cannot be taken since all of V lies on level 1. Each node \(v\in V\) is connected to all of \(\mathcal {I}(B_v)\), which lie on level 2. Only s remains, which is connected to \(\tilde{p}\)—the first investigated node on level 2.

References

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Authors and Affiliations

  1. Theoretical Computer Science, Osnabrück University, Osnabrück, Germany

    Markus Chimani, Ivo Hedtke & Tilo Wiedera

Authors
  1. Markus Chimani
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  2. Ivo Hedtke
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  3. Tilo Wiedera
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Corresponding author

Correspondence to Ivo Hedtke .

Editor information

Editors and Affiliations

  1. University of Helsinki, Helsinki, Finland

    Veli Mäkinen

  2. University of Helsinki, Helsinki, Finland

    Simon J. Puglisi

  3. University of Helsinki, Helsinki, Finland

    Leena Salmela

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Chimani, M., Hedtke, I., Wiedera, T. (2016). Limits of Greedy Approximation Algorithms for the Maximum Planar Subgraph Problem. In: Mäkinen, V., Puglisi, S., Salmela, L. (eds) Combinatorial Algorithms. IWOCA 2016. Lecture Notes in Computer Science(), vol 9843. Springer, Cham. https://doi.org/10.1007/978-3-319-44543-4_26

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  • DOI: https://doi.org/10.1007/978-3-319-44543-4_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-44542-7

  • Online ISBN: 978-3-319-44543-4

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