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An algorithm and upper bounds for the weighted maximal planar graph problem

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Journal of the Operational Research Society

Abstract

In this paper, we investigate the weighted maximal planar graph (WMPG) problem. Given a complete, edge-weighted, simple graph, the WMPG problem involves finding a subgraph with the highest sum of edge weights that is maximal planar, namely, it can be embedded in the plane without any of its edges intersecting, and no additional edge can be added to the subgraph without violating its planarity. We present a new integer linear programming (ILP) model for this problem. We then develop a cutting-plane algorithm to solve the WMPG problem based on the proposed ILP model. This algorithm enables the problem to be solved more efficiently than previously reported algorithms. New upper bounds are also provided, which are useful in evaluating the quality of heuristic solutions or in generating initial solutions for meta-heuristics. Computational results are reported for a set of 417 test instances of size varying from 6 to 100 nodes including 105 instances from the literature and 312 randomly generated instances. The computational results indicate that instances with up to 24 nodes can be solved optimally in reasonable computational time and the new upper bounds for larger instances significantly improve existing upper bounds.

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References

  • Ahmadi-Javid A and Ramshe N (2013). On the block layout shortest loop design problem. IIE Transactions 45 (5): 494–501.

    Article  Google Scholar 

  • Bondy JA and Murty USR (1976). Graph Theory with Applications. Palgrave Macmillan: London.

    Book  Google Scholar 

  • de Fraysseix H, Pach J and Pollack R (1990). How to draw a planar graph on a grid. Combinatorica 10 (1): 41–51.

    Article  Google Scholar 

  • Domschke W and Drexl A (1985). Location and Layout Planning: An International Bibliography. Springer-Verlag: New York.

    Book  Google Scholar 

  • Dyer ME, Foulds LR and Frieze AM (1985). Analysis of heuristics for finding a maximum weight planar subgraph. European Journal of Operational Research 20 (1): 102–114.

    Article  Google Scholar 

  • Eades P, Foulds LR and Giffin JW (1982). An efficient heuristic for identifying a max-weight planar subgraph, Lecture Notes in Mathematics, 952. Springer-Verlag: Berlin.

  • Farahani ZR, Laporte G and Sharifyazdi M (2005). A practical exact algorithm for the shortest loop design problem in a block layout. International Journal of Production Research 43 (9): 1879–1887.

    Article  Google Scholar 

  • Foulds LR (1983). Techniques for facilities layout: Deciding which pairs of activities should be adjacent. Management Science 29 (12): 1414–1426.

    Article  Google Scholar 

  • Foulds LR (1992). Graph Theory Applications. Springer-Verlag: New York.

    Book  Google Scholar 

  • Foulds LR (1993a). Graph theoretic-based decision support for facilities layout. IFIP Transactions B: Applications in Technology 11: 145–157.

    Google Scholar 

  • Foulds LR (1993b). Layout manager: A decision support system for facilities planning. IFIP Transactions B: Applications in Technology 11: 293–300.

    Google Scholar 

  • Foulds LR (1997). LayoutManager: A micro-computer based decision support system for facilities planning. Decision Support Systems 20 (3): 199–213.

    Article  Google Scholar 

  • Foulds LR and Giffin JW (1987). Facilities layout generalized model solved by n-boundary shortest path heuristics. European Journal of Operational Research 28 (3): 382–391.

    Article  Google Scholar 

  • Foulds LR and Giffin JW (1990). Some graph theoretic approaches to certain facilities layout models. Annals of the New York Academy of Sciences 576: 176–191.

    Google Scholar 

  • Foulds LR and Robinson DF (1976). A strategy for solving the plant layout problem. Operational Research Quarterly 27 (4): 845–885.

    Article  Google Scholar 

  • Foulds LR and Robinson DF (1978). Graph theoretic heuristics for the plant layout problem. International Journal of Production Research 16 (1): 27–37.

    Article  Google Scholar 

  • Foulds LR and Tran HV (1986). Library layout via graph theory. Computers & Industrial Engineering 10 (3): 245–252.

    Article  Google Scholar 

  • Foulds LR, Gibbons PB and Giffin JW (1985). Facilities layout adjacency determination: An experimental comparison of three graph theoretic heuristics. Operations Research 33 (5): 1091–1106.

    Article  Google Scholar 

  • Garey MR and Johnson DS (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman: New York.

    Google Scholar 

  • Giffin JW (1984). Graph theory techniques for facilities layout. PhD Thesis, University of Canterbury, New Zealand.

  • Giffin JW and Foulds LR (1987). Facility layout generalized model solved by n-boundary shortest path heuristics. European Journal of Operational Research 28 (3): 109–116.

    Article  Google Scholar 

  • Giffin JW, Watson KH and Foulds LR (1995). Orthogonal layouts using the deltahedron heuristic. Australasian Journal of Combinatorics 12: 127–144.

    Google Scholar 

  • Hales HL (1984). Computer-Aided Facilities Planning. M. Dekker: New York.

    Google Scholar 

  • Hasan M and Osman IH (1995). Local search algorithms for the maximal planar layout problem. International Transactions in Operational Research 2 (1): 89–106.

    Article  Google Scholar 

  • Hassan MMD and Hogg GL (1987). A review of graph theory application to the facilities layout problem. Omega 15 (4): 291–300.

    Article  Google Scholar 

  • Hopcroft J and Tarjan R (1974). Efficient planarity testing. Journal of the ACM 21 (4): 549–568.

    Article  Google Scholar 

  • Kusiak A and Heragu SS (1987). The facility layout problem. European Journal of Operational Research 29 (3): 229–251.

    Article  Google Scholar 

  • Lempel A, Even S and Cederbaum I (1967). An algorithm for planarity testing of graphs. In: Rosenstiehl P (ed). Theory of Graphs. Gordon and Breach: New York, pp 215–232.

    Google Scholar 

  • Mehlhorn K (1984). Data Structures and Algorithms II: Graph Algorithms and NP-Completeness. Springer-Verlag: Berlin.

    Book  Google Scholar 

  • Mehlhorn K and Mutzel P (1996). On the embedding phase of the Hopcroft and Tarjan planarity testing algorithm. Algorithmica 16 (2): 233–242.

    Article  Google Scholar 

  • Meller RD and Gau KY (1996). The facility layout problem: Recent and emerging trends and perspectives. Journal of Manufacturing Systems 15 (5): 351–366.

    Article  Google Scholar 

  • Merker J and Wascher G (1997). Two new heuristic algorithms for the maximal planar layout problem. OR Spectrum 19 (2): 131–137.

    Article  Google Scholar 

  • Montreuil B and Ratliff D (1989). Utilizing cut trees as design skeletons for facility layout. IIE Transactions 21 (2): 136–143.

    Article  Google Scholar 

  • Montreuil B, Ratliff D and Goetschalckx M (1987). Matching based interactive facility layout. IIE Transactions 19 (3): 271–279.

    Article  Google Scholar 

  • Muther R (1955). Practical Plant Layout. McGraw-Hill: New York.

    Google Scholar 

  • Osman IH (2006). A tabu search procedure based on a random roulette diversification for the weighted maximal planar graph problem. Computers & Operations Research 33 (9): 2526–2546.

    Article  Google Scholar 

  • Osman IH, Hasan M and Abdullah A (2002). Linear programming based meta-heuristics for the weighted maximal planar graph. Journal of the Operational Research Society 53 (10): 1142–1149.

    Article  Google Scholar 

  • Osman IH, Al-Ayoubi B and Barake M (2003). A greedy random adaptive search procedure for the weighted maximal planar graph problem. Computers & Industrial Engineering 45 (4): 635–651.

    Article  Google Scholar 

  • Schnyder W (1990). Embedding planar graphs on the grid, in Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM: Philadelphia, pp. 138–148.

    Google Scholar 

  • Seppanen JJ and Moore JM (1970). Facilities planning with graph theory. Management Science 17 (4): 242–253.

    Article  Google Scholar 

  • Seppanen JJ and Moore JM (1975). String processing algorithms for plant layout problems. International Journal of Production Research 13 (3): 239–254.

    Article  Google Scholar 

  • Singh SP and Sharma RRK (2006). A review of different approaches to the facility layout problems. International Journal of Advanced Manufacturing Technology 30 (5–6): 425–433.

    Article  Google Scholar 

  • Tompkins JA, Bozer YA, Tanchoco JMA and White JA (2003). Facilities Planning. John Wiley & Sons: New York.

    Google Scholar 

  • Wascher G and Merker J (1997). A comparative evaluation of heuristics for the adjacency problem in facility layout planning. International Journal of Production Research 35 (2): 447–466.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for the valuable comments which helped improve the paper. The authors would like to thank Prof. Ibrahim H. Osman, from the American University of Beirut, who helped us by graciously sending us his test problems and giving his constructive advice.

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Authors and Affiliations

  1. Amirkabir University of Technology, Tehran, Iran

    Amir Ahmadi-Javid

  2. GERAD, HEC Montréal, Montréal, Canada

    Amir Ardestani-Jaafari

  3. Instituto de Informática, Universidade Federal de Goiás, Goiania, Brasil

    Leslie R Foulds

  4. CIRRELT and Départment d’informatique et de recherche opérationnelle, Université de Montréal, Montréal, Canada

    Hossein Hojabri

  5. Kingston University, London, UK

    Reza Zanjirani Farahani

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  1. Amir Ahmadi-Javid
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  2. Amir Ardestani-Jaafari
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  3. Leslie R Foulds
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  4. Hossein Hojabri
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  5. Reza Zanjirani Farahani
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Ahmadi-Javid, A., Ardestani-Jaafari, A., Foulds, L. et al. An algorithm and upper bounds for the weighted maximal planar graph problem. J Oper Res Soc 66, 1399–1412 (2015). https://doi.org/10.1057/jors.2014.98

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  • DOI: https://doi.org/10.1057/jors.2014.98

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