Approximation Algorithms for Facial Cycles in Planar Embeddings

Authors Giordano Da Lozzo , Ignaz Rutter

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Giordano Da Lozzo
  • Computer Science Department, Roma Tre University, Italy
Ignaz Rutter
  • Department of Computer Science and Mathematics, University of Passau, Germany

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Giordano Da Lozzo and Ignaz Rutter. Approximation Algorithms for Facial Cycles in Planar Embeddings. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Consider the following combinatorial problem: Given a planar graph G and a set of simple cycles C in G, find a planar embedding E of G such that the number of cycles in C that bound a face in E is maximized. This problem, called Max Facial C-Cycles, was first studied by Mutzel and Weiskircher [IPCO '99, and then proved NP-hard by Woeginger [Oper. Res. Lett., 2002,]. We establish a tight border of tractability for Max Facial C-Cycles in biconnected planar graphs by giving conditions under which the problem is NP-hard and showing that strengthening any of these conditions makes the problem polynomial-time solvable. Our main results are approximation algorithms for Max Facial C-Cycles. Namely, we give a 2-approximation for series-parallel graphs and a (4+epsilon)-approximation for biconnected planar graphs. Remarkably, this provides one of the first approximation algorithms for constrained embedding problems.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Planar Embeddings
  • Facial Cycles
  • Complexity
  • Approximation Algorithms


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