# Heuristics for the Maximum Outerplanar Subgraph Problem

- 58 Downloads
- 1 Citations

## Abstract

Determining the maximum outerplanar subgraph of a given graph is known to be an NP-complete problem. In the literature there are no earlier experiment on approximating the maximum outerplanar subgraph problem. In this paper we compare solution quality and running times of different heuristics for finding maximum outerplanar subgraphs. We compare a greedy heuristic against a triangular cactus heuristic and its greedy variation. We also use the solutions from the greedy heuristics as initial solutions for a simulated annealing algorithm.

The main experimental result is that simulated annealing with initial solution taken from the greedy triangular cactus heuristic yields the best known approximations for the maximum outerplanar subgraph problem.

## Key Words

triangular cactus heuristic simulated annealing maximum outerplanar subgraph problem## Preview

Unable to display preview. Download preview PDF.

## References

- Aarts, E. and J. Lenstra. (1997).
*Local Search in Combinatorial Optimization*. John Wiley and Sons.Google Scholar - Aarts, E. and P. van Laarhoven. (1985). “Statistical Cooling: A General Approach to Combinatorial Optimization Problems.”
*Philips J. Res.*40, 193–226.Google Scholar - Aragon, C.R., D.S. Johnson, L.A. McGeoch, and C. Schevon. (1991). “Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning.”
*Oper. Res.*3(39), 378–406.Google Scholar - Boyer, J. and W. Myrvold. (1999). “Stop Minding Your P’s and Q’s: A Simplified
*O*(*n*) Planar Embedding Algorithm.” In*Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms*. pp. 140–146.Google Scholar - Brehaut, W. (1977). “An Efficient Outerplanarity Algorithm.” In
*Proceedings of the 8th South-Eastern Conference on Combinatorics, Graph Theory, and Computing*. pp. 99–113.Google Scholar - Călinescu, G., C. Fernandes, U. Finkler, and H. Karloff. (1998). “A Better Approximation Algorithm for Finding Planar Subgraphs.”
*J. Algorithms*27(2), 269–302.CrossRefGoogle Scholar - Călinescu, G., C. Fernandes, H. Karloff, and A. Zelikovsky. (2003). “A New Approximation Algorithm for Finding Heavy Planar Subgraphs.”
*Algorithmica*36(2), 179–205.CrossRefGoogle Scholar - Cimikowski, R. (1995a). “An Analysis of Heuristics for the Maximum Planar Subgraph Problem.” In
*Proceedings of the 6th ACM-SIAM Symposium on Discrete Algorithms*. pp. 322–331.Google Scholar - Cimikowski, R. (1997). “An Analysis of Heuristics for Graph Planarization.”
*J. Inf. Opt. Sci.*18(1), 49–73.MathSciNetGoogle Scholar - Cimikowski, R. and D. Coppersmith. (1996). “The Sizes of Maximal Planar, Outerplanar, and Bipartite Planar Subgraphs.”
*Discr. Math.*149, 303–309.CrossRefGoogle Scholar - Felsner, S., G. Liotta, and S. Wismath. (2002). “Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions.” In
*Proceedings of Graph Drawing: 9th International Symposium (GD’01)*, vol. 2265 of*LNCS*. pp. 328–342.Google Scholar - Galil, Z., G. Italiano, and N. Sarnak. (1999). “Fully Dynamic Planarity Testing with Applications.”
*J. ACM*46(1), 28–91.CrossRefGoogle Scholar - Garey, M. and D. Johnson. (1979)
*Computers and Intractability. A Guide to the Theory of NP-Completeness*. W.H. Freeman.Google Scholar - Glauber, R. (1963). “Time-Dependent Statistics of the Ising Model.”
*Math. Phys.*4(2), 294–307.CrossRefGoogle Scholar - Goldschmidt, O. and A. Takvorian. (1994). “An Efficient Graph Planarization Two-Phase Heuristic.”
*Networks*24, 69–73.Google Scholar - Guy, R. (1974).
*Combinatorics, London Math. Soc. Lecture Notes 13*, Chapt. Outerthickness and outercoarseness of graphs, Cambridge University Press, pp. 57–60.Google Scholar - Harary, F. (1971).
*Graph Theory*. Addison-Wesley.Google Scholar - Hopcroft, J. and R. Tarjan. (1974). “Efficient Planarity Testing.”
*J. ACM*21, 549–568.CrossRefGoogle Scholar - Johnson, D. (2002). “A Theoretician’s Guide to the Experimental Analysis of Algorithms.” In
*Data Structures, Near Neighbor Searches, and Methodology: Fifth and Sixth DIMACS Implementation Challenges*. pp. 215–250.Google Scholar - Johnson, D., C.R. Aragon, L.A. McGeoch, and C. Schevon. (1989). “Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning.”
*Oper. Res.*6(37), 865–892.Google Scholar - Kant, G. (1992). “An
*O*(*n*^{2}) Maximal Planarization Algorithm Based on PQ-Trees.” Technical Report, Utrecht University. Technical Report RUU-CS-92-03.Google Scholar - Kant, G. (1996). “Augmenting Auterplanar Graphs.”
*J. Algorithms*21, 1–25.CrossRefGoogle Scholar - Kirkpatrick, S., C. Gelatt, and M. Vecchi. (1983). “Optimization by Simulated Annealing.”
*Science*220, 671–680.Google Scholar - LEDA. (2003). “LEDA Version 4.3 (commercial).” Available at http://www.algorithmic-solutions.com.
- Liebers, A. (2001). “Planarizing Graphs—A Survey and Annotated Bibliography.”
*J. Graph Alg. and Appl.*5(1), 1–74.Google Scholar - Liu, P. and R. Geldmacher. (1977). “On the Deletion of Nonplanar Edges of a Graph.” In
*Proceedings of the 10th South-Eastern Conference on Combinatorics, Graph Theory, and Computing*. pp. 727–738.Google Scholar - Maheshwari, A. and N. Zeh. (1999). “External Memory Algorithms for Outerplanar Graphs.” In
*Proceedings of the 10th International Symposium on Algorithms and Computations*, Vol. 1741 of*LNCS*. pp. 307–316.Google Scholar - Manning, J. and M. Atallah. (1992). “Fast Detection and Display of Symmetry in Outerplanar Graphs.”
*Discr. Appl. Math.*39(1), 13–35.CrossRefGoogle Scholar - Metropolis, N., A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller. (1953). “Equation of State Calculation by Fast Computing Machines.”
*J. Chem. Phys.*21, 1087–1091.CrossRefGoogle Scholar - Mitchell, S. (1979). “Linear Algorithms to Recognize Outerplanar and Maximal Outerplanar Graphs.”
*Inf. Proc. Lett.*9(5), 177–189.Google Scholar - Mutzel, P., T. Odenthal, and M. Scharbrodt. (1998). “The Thickness of Graphs: A Survey.”
*Graphs Comb.*14, 59–73.Google Scholar - Poranen, T. (2003). “Apptopinv—User’s guide.” Technical Report A-2003-3, University of Tampere, Department of Computer Sciences.Google Scholar
- Poranen, T. (2004). “A Simulated Annealing Algorithm for Determining Maximum Planar Subgraphs.”
*Int. J. Comput. Math.*81(5), 555 – 568.CrossRefGoogle Scholar - Reeves, C. (ed.). (1995).
*Modern Heuristic Techniques for Combinatorial Problems*. McGraw-Hill.Google Scholar - Resende, M. and C. Ribeiro. (1997). “A GRASP for Graph Planarization.”
*Networks*29, 173–189.CrossRefGoogle Scholar - Shih, W.-K. and W.-L. Hsu. (1999). “A New Planarity Test.”
*Theor. Comput. Sci.*223, 179–191.CrossRefGoogle Scholar - Syslo, M. (1978). “Outerplanar Graphs: Characterizations, Testing, Coding and Counting.”
*Bull. Acad. Polon. Sci. Sèr. Sci. Math. Astronom. Phys.*26(8), 675–684.Google Scholar - Syslo, M. and M. Iri. (1979). “Efficient Outerplanarity Testing.”
*Fund. Inf.*II, 261–275.Google Scholar - van Laarhoven, P. and E. Aarts. (1987).
*Simulated Annealing: Theory and Applications.*Kluwer.Google Scholar - Vrtò, I. (2002). “Crossing Numbers of Graphs: A Bibliography.” Available at ftp://ifi.savba.sk/pub/imrich/crobib.ps.gz.
- Wiegers, M. (1984). “Recognizing Outerplanar Graphs in Linear Time.” In
*Graph-Theoretic Concepts in Computer Science, International Workshop WG’86*, Vol. 246 of*LNCS*. pp. 165–176.Google Scholar - Yannakakis, M. (1978). “Node- and Edge-Deletion NP-Complete Problems.” In
*Proceedings of the 10th Annual ACM Symposium on Theory of Computing*. pp. 253–264.Google Scholar