Abstract.
We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms.
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Agarwal, P.K., Sharir, M., Shor, P.: Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences. J. Comb. Theory, Ser. A 52, 228–274 (1989)
Ahuja, R.K., Hochbaum, D.S., Orlin, J.B.: A cut-based algorithm for the nonlinear dual of the minimum cost network flow problem, manuscript available at http://web.mit.edu/jorlin/www/working/_papers.html
Ahuja, R.K., Hochbaum, D.S., Orlin, J.B.: Solving the convex cost integer dual network flow problem, G. Cornuejols, R.E. Burkard and G.J. Woeginger, eds., Proc. IPCO'99, Lect. Notes Comput. Sci. 1610 (Springer, 1999) 31–44
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry. (Springer, 1997)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. (Freeman, 1979)
Hartmann, S.: Packing problem and project scheduling models: an integrating perspective. J. Oper. Res. Soc. 51, 1083–1092 (2000)
Imahori, S.: Local search heuristics for the rectangle packing problem with general spatial costs. Master thesis, Department of Applied Mathematics and Physics, Graduate School of Informatics. Kyoto University, 2001
Johnson, D.S.: Local optimization and the traveling salesman problem, M.S. Peterson, ed., Automata, Languages and Programming, Lect. Notes Comput. Sci. 443, Springer, 1990, pp. 446–461
Liu, D., Teng, H.: An improved BL-algorithm for genetic algorithm of the orthogonal packing of rectangles. Eur. J. Oper. Res. 112, 413–420 (1999)
Lodi, A., Martello, S., Vigo, D.: Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems. Informs J. Comput. 11, 345–357 (1999)
Murata, H., Fujiyoshi, K., Nakatake, S., Kajitani, Y.: VLSI module placement based on rectangle-packing by the sequence-pair. IEEE Trans. Comput. Aided Des. 15–12, 1518–1524 (1996)
Nakatake, S., Fujiyoshi, K., Murata, H., Kajitani, Y.: Module placement on BSG-structure and IC layout applications. Proc. Intl. Conf. Comput. Aided Des. 484–491 (1996)
Nonobe, K., Ibaraki, T.: Formulation and tabu search algorithm for the resource constrained project scheduling problem. Essays and Surveys in Metaheuristics. (Kluwer Academic Publishers, 2001) 557–588
Selman, B., Kautz, H.A., Cohen, B.: Noise strategies for improving local search. Proc. 12th National Conf. Artif. Intell. 1994, pp. 337–343
Takahashi, T.: An algorithm for finding a maximum-weight decreasing sequence in a permutation, motivated by rectangle packing problem (in Japanese), Technical Report of IEICE VLD96-30, 1996, pp. 31–35
Tang, X., Tian, R., Wong, D.F.: Fast evaluation of sequence pair in block placement by longest common subsequence computation. Proc. Des. Autom. Test Eur. Conf. 2000, pp. 106–111
Tang, X., Wong, D.F.: Fast-SP: a fast algorithm for block placement based on sequence pair. Proc. Asia S. Pac. Des. Autom. Conf. 2001, pp. 521–526
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Key words. rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristics
Mathematics Subject Classification (1991): 20E28, 20G40, 20C20
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Imahori, S., Yagiura, M. & Ibaraki, T. Local search algorithms for the rectangle packing problem with general spatial costs. Math. Program., Ser. B 97, 543–569 (2003). https://doi.org/10.1007/s10107-003-0427-1
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DOI: https://doi.org/10.1007/s10107-003-0427-1