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Approximation Algorithms for Facial Cycles in Planar Embeddings

Authors Giordano Da Lozzo , Ignaz Rutter

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

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Abstract

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, http://dx.doi.org/10.1007/3-540-48777-8_27) and then proved NP-hard by Woeginger [Oper. Res. Lett., 2002, http://dx.doi.org/10.1016/S0167-6377(02)00119-0].
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.

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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) https://doi.org/10.4230/LIPIcs.ISAAC.2018.41

Author Details

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

References

  1. Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, and Ignaz Rutter. Testing Planarity of Partially Embedded Graphs. ACM Trans. Alg., 11(4):32:1-32:42, 2015. URL: http://dx.doi.org/10.1145/2629341.
  2. Patrizio Angelini, Giuseppe Di Battista, and Maurizio Patrignani. Finding a Minimum-Depth Embedding of a Planar Graph in O(n⁴) Time. Algorithmica, 60:890-937, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9380-6.
  3. Brenda S. Baker. Approximation Algorithms for NP-complete Problems on Planar Graphs. J. ACM, 41(1):153-180, January 1994. URL: http://dx.doi.org/10.1145/174644.174650.
  4. Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inflexible edges. Comput. Geom., 55:26-40, 2016. URL: http://dx.doi.org/10.1016/j.comgeo.2016.03.001.
  5. Thomas Bläsius, Ignaz Rutter, and Dorothea Wagner. Optimal Orthogonal Graph Drawing with Convex Bend Costs. ACM Trans. Algorithms, 12(3):33:1-33:32, 2016. Google Scholar
  6. Giordano Da Lozzo, Vít Jelínek, Jan Kratochvíl, and Ignaz Rutter. Planar Embeddings with Small and Uniform Faces. In ISAAC'14, volume 8889 of LNCS, pages 633-645. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13075-0.
  7. Giordano Da Lozzo and Ignaz Rutter. On the complexity of realizing facial cycles. CoRR, abs/1607.02347, 2016. URL: http://arxiv.org/abs/1607.02347.
  8. Giuseppe Di Battista and Fabrizio Frati. Drawing Trees, Outerplanar Graphs, Series-Parallel Graphs, and Planar Graphs in a Small Area. In Thirty Essays on Geometric Graph Theory, pages 121-165. Springer New York, 2013. URL: http://dx.doi.org/10.1007/978-1-4614-0110-0_9.
  9. Giuseppe Di Battista and Roberto Tamassia. On-Line Graph Algorithms with SPQR-Trees. In Michael S. Paterson, editor, ICALP'90, volume 443 of LNCS, pages 598-611. Springer, 1990. URL: http://dx.doi.org/10.1007/BFb0032061.
  10. Christoph Dornheim. Planar Graphs with Topological Constraints. JGAA, 6(1):27-66, 2002. URL: http://dx.doi.org/10.7155/jgaa.00044.
  11. M. R. Garey, David S. Johnson, and Robert Endre Tarjan. The Planar Hamiltonian Circuit Problem is NP-Complete. SIAM J. Comput., 5(4):704-714, 1976. URL: http://dx.doi.org/10.1137/0205049.
  12. Ashim Garg and Roberto Tamassia. On the Computational Complexity of Upward and Rectilinear Planarity Testing. SIAM J. on Comput., 31(2):601-625, 2001. URL: http://dx.doi.org/10.1137/S0097539794277123.
  13. Carsten Gutwenger and Petra Mutzel. A Linear Time Implementation of SPQR-trees. In Joe Marks, editor, GD'00, volume 1984 of LNCS, pages 77-90. Springer, 2001. URL: http://dx.doi.org/10.1007/3-540-44541-2_8.
  14. Joseph Douglas Horton and Kyriakos Kilakos. Minimum Edge Dominating Sets. SIAM J. Discrete Math., 6(3):375-387, 1993. URL: http://dx.doi.org/10.1137/0406030.
  15. Frank Kammer. Determining the Smallest k Such That G Is k-Outerplanar. In ESA'07, volume 4698 of LNCS, pages 359-370. Springer, 2007. URL: http://dx.doi.org/10.1007/978-3-540-75520-3_33.
  16. Petra Mutzel and René Weiskircher. Optimizing over All Combinatorial Embeddings of a Planar Graph. In IPCO'99, volume 1610 of LNCS, pages 361-376. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-48777-8_27.
  17. Roberto Tamassia. On Embedding a Graph in the Grid with the Minimum Number of Bends. SIAM J. Comput., 16(3):421-444, 1987. URL: http://dx.doi.org/10.1137/0216030.
  18. Hassler Whitney. Congruent Graphs and the Connectivity of Graphs. American Journal of Mathematics, 54(1):150-168, 1932. URL: http://www.jstor.org/stable/2371086.
  19. Gerhard J. Woeginger. Embeddings of planar graphs that minimize the number of long-face cycles. Oper. Res. Lett., 30(3):167-168, 2002. URL: http://dx.doi.org/10.1016/S0167-6377(02)00119-0.
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