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A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs

Authors Parinya Chalermsook, Andreas Schmid, Sumedha Uniyal

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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Graph Drawing
  • Matroid Matching
  • Maximum Planar Subgraph
  • Local Search Algorithms

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Abstract

A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph G contains a cactus subgraph C where C contains at least a 1/6 fraction of the triangular faces of G. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing "dense planar structures" inside any graph: (i) A 1/6 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous 1/11-approximation; (ii) An alternate (and arguably more illustrative) proof of the 4/9 approximation algorithm for finding a planar subgraph with a maximum number of edges.
Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind.

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Parinya Chalermsook, Andreas Schmid, and Sumedha Uniyal. A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.STACS.2019.19

Author Details

Parinya Chalermsook
  • Aalto University, Espoo, Finland
Andreas Schmid
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Sumedha Uniyal
  • Aalto University, Espoo, Finland

Funding

  • Chalermsook, Parinya: Part of this work was done while PC and AS were visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant #CCF-1740425. Parinya has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 759557) and by Academy of Finland Research Fellows, under grant number 310415 and 314284.
  • Uniyal, Sumedha: Partially supported by Academy of Finland under the grant agreement number 314284.

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