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On three soft rectangle packing problems with guillotine constraints

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Abstract

We investigate how to partition a rectangular region of length \(L_1\) and height \(L_2\) into n rectangles of given areas \((a_1, \dots , a_n)\) using two-stage guillotine cuts, so as to minimize either (i) the sum of the perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of the rectangles. These problems play an important role in the ongoing Vietnamese land-allocation reform, as well as in the optimization of matrix multiplication algorithms. We show that the first problem can be solved to optimality in \({{\mathcal {O}}}(n \log n)\), while the two others are NP-hard. We propose mixed integer linear programming formulations and a binary search-based approach for solving the NP-hard problems. Experimental analyses are conducted to compare the solution approaches in terms of computational efficiency and solution quality, for different objectives.

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References

  1. Arimoto, Y.: Impact of land readjustment project on farmland use and structural adjustment: the case of Niigata, Japan. In: Agricultural and Applied Economics Association 2010 AAEA, CAES, WAEA Joint Annual Meeting, Denver, USA (2010)

  2. Beaumont, O., Boudet, V., Rastello, F., Robert, Y.: Matrix multiplication on heterogeneous platforms. IEEE Trans. Parallel Distrib. Syst. 12(10), 1033–1051 (2001)

    Article  Google Scholar 

  3. Beaumont, O., Boudet, V., Rastello, F., Robert, Y.: Partitioning a square into rectangles: NP-completeness and approximation algorithms. Algorithmica 34(3), 217–239 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borgwardt, S., Brieden, A., Gritzmann, P.: Constrained minimum-k-star clustering and its application to the consolidation of farmland. Oper. Res. 11(1), 1–17 (2011)

    MATH  Google Scholar 

  5. Borgwardt, S., Brieden, A., Gritzmann, P.: Geometric clustering for the consolidation of farmland and woodland. Math. Intell 36(2), 37–44 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brieden, A., Gritzmann, P.: A quadratic optimization model for the consolidation of farmland by means of lend-lease agreements. In: Ahr, D., Fahrion, R., Oswald, M., Reinelt, G. (eds.) Operations Research Proceedings 2003, pp. 324–331. Springer, Berlin, Heidelberg (2004)

    Chapter  Google Scholar 

  7. Bui, Q.T., Pham, Q.D., Deville, Y.: Solving the agricultural land allocation problem by constraint-based local search. In: Schulte, C. (ed.) Principles and Practice of Constraint Programming. Lecture Notes in Computer Science, vol. 8124, pp. 749–757. Springer, Berlin, Heidelberg (2013)

    Chapter  Google Scholar 

  8. Cay, T., Uyan, M.: Evaluation of reallocation criteria in land consolidation studies using the analytic hierarchy process (AHP). Land Use Policy 30(1), 541–548 (2013)

    Article  Google Scholar 

  9. Cay, T., Ayten, T., Iscan, F.: An investigation of reallocation model based on interview in land consolidation. In: Proceeding of the 23rd FIG Congress, Shaping the Change, Munich, Germany (2006)

  10. Cay, T., Ayten, T., Iscan, F.: Effects of different land reallocation models on the success of land consolidation projects: social and economic approaches. Land Use Policy 27(2), 262–269 (2010)

    Article  Google Scholar 

  11. Demetriou, D., See, L., Stillwell, J.: A spatial genetic algorithm for automating land partitioning. Int. J. Geogr. Inf. Sci. 27, 2391–2409 (2013)

    Article  Google Scholar 

  12. Fügenschuh, A., Junosza-Szaniawski, K., Lonc, Z.: Exact and approximation algorithms for a soft rectangle packing problem. Optim. J. Math. Program. Oper. Res. 63(11), 1637–1663 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)

    MATH  Google Scholar 

  14. General Statistical Office: Statistical Yearbook 2004. Statistical Publishing House, Hanoi (2004)

    Google Scholar 

  15. Gliesch, A., Ritt, M., Moreira, MCO.: A genetic algorithm for fair land allocation. In: Proceedings of the Genetic and Evolutionary Computation Conference, ACM, New York, NY, USA, GECCO ’17, pp. 793–800 (2017)

  16. Hakli, H., Uuz, H.: A novel approach for automated land partitioning using genetic algorithm. Expert Syst. Appl. 82(1), 10–18 (2017)

    Article  Google Scholar 

  17. Heltberg, R.: Rural market imperfections and the farm size-productivity relationship: evidence from Pakistan. World Dev. 26, 1807–1826 (1998)

    Article  Google Scholar 

  18. Huang, Q., Li, M., Chen, Z., Li, F.: Land consolidation: an approach for sustainable development in rural China. AMBIO 40(1), 93–95 (2011)

    Article  Google Scholar 

  19. Ibaraki, T., Nakamura, K.: Packing problems with soft rectangles. In: Almeida, F., Blesa Aguilera, M.J., Blum, C., Moreno Vega, J.M., Pérez Pérez, M., Roli, A., Sampels, M. (eds.) Hybrid Metaheuristics, pp. 13–27. Springer, Berlin, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Ji, P., He, K., Jin, Y., Lan, H., Li, C.: An iterative merging algorithm for soft rectangle packing and its extension for application of fixed-outline floorplanning of soft modules. Comput. Oper. Res. 86, 110–123 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Rural Econ. (2008) Agricultural mechanization: When does it go over the start? http://ipsard.gov.vn/mobile/tID2264_Co-gioi-hoa-nong-nghiep-Khi-nao-qua-buoc-khoi-dong-.html. Accessed 01 May 2018

  22. Kong, T.Y., Mount, D.M., Werman, M.: The decomposition of a square into rectangles of minimal perimeter. Discrete Appl. Math. 16(3), 239–243 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kong, T.Y., Mount, D.M., Werman, M.: The decomposition of a rectangle into rectangles of minimal perimeter. SIAM J. Comput. 17(6), 1215–1231 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lam, M.L.: Land fragmentation-a constraint for Vietnam agriculture. Vietnam Socio Econ. Dev. 26, 73–80 (2001)

    Google Scholar 

  25. March, S.P., MacAulay, T.G.: Farm size and land use change in Vietnam following land reforms. In: 47th Annual Conference of the Australian Agricultural and Resource Economics Society, Fremantle, Australia (2006)

  26. Nagamochi, H., Abe, Y.: An approximation algorithm for dissecting a rectangle into rectangles with specified areas. Discrete Appl. Math. 155(4), 523–537 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Paes, F., Pessoa, A., Vidal, T.: A hybrid genetic algorithm with decomposition phases for the unequal area facility layout problem. Eur. J. Oper. Res. 256(3), 742–756 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pham, V.H., MacAulay, G.T., Marsh, S.P.: The economics of land fragmentation in the north of Vietnam. Aust. J. Agric. Resour. Econ. 51(2), 195–211 (2007)

    Article  Google Scholar 

  29. Sundqvist, P., Anderson, L.: A study of the impacts of land fragmentation on agricultural productivity in Northern Vietnam. Bachelor’s thesis, Uppsala University, Sweden (2006)

  30. Wilber, R.: The concave least-weight subsequence problem revisited. J. Algorithms 9(3), 418–425 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. Young, F.Y., Chu, C.C.N., Luk, W.S., Wong, Y.C.: Handling soft modules in general nonslicing floorplan using lagrangian relaxation. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 20(5), 687–692 (2001)

    Article  Google Scholar 

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Acknowledgements

This research has been partly funded by the the National Council for Scientific and Technological Development (CNPQ–grant number 308498/2015-1) and FAPERJ in Brazil (grant number E-26/203.310/2016). This support is gratefully acknowledged.

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Authors and Affiliations

  1. Daily-Opt Joint Stock Company, Hanoi, Vietnam

    Quoc Trung Bui

  2. FPT University, Hanoi, Vietnam

    Quoc Trung Bui

  3. Risk division, Vietnam Technological and Commercial Joint Stock Bank, Hanoi, Vietnam

    Quoc Trung Bui

  4. Departamento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil

    Thibaut Vidal

  5. ORLab, University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam

    Minh Hoàng Hà

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  1. Quoc Trung Bui
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  2. Thibaut Vidal
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  3. Minh Hoàng Hà
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Correspondence to Minh Hoàng Hà.

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Bui, Q.T., Vidal, T. & Hà, M.H. On three soft rectangle packing problems with guillotine constraints. J Glob Optim 74, 45–62 (2019). https://doi.org/10.1007/s10898-019-00741-w

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