Mathematical Programming

, Volume 102, Issue 3, pp 519–530 | Cite as

The two-dimensional cutting stock problem revisited

  • Steven S. Seiden
  • Gerhard J. Woeginger


In the strip packing problem (a standard version of the two-dimensional cutting stock problem), the goal is to pack a given set of rectangles into a vertical strip of unit width so as to minimize the total height of the strip needed. The k-stage Guillotine packings form a particularly simple and attractive family of feasible solutions for strip packing. We present a complete analysis of the quality of k-stage Guillotine strip packings versus globally optimal packings: k=2 stages cannot guarantee any bounded asymptotic performance ratio. k=3 stages lead to asymptotic performance ratios arbitrarily close to 1.69103; this bound is tight. Finally, k=4 stages yield asymptotic performance ratios arbitrarily close to 1.


Cutting stock Strip packing Guillotine cuts Packing problem Approximation scheme Worst case analysis 


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  1. 1.
    Baker, B.S., Brown, D.J., Katseff, H.P.: A 5/4 algorithm for two-dimensional bin packing. J. Alg. 2, 348–368 (1981)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Baker, B.S., Coffman, E.G., Rivest, R.L.: Orthogonal packings in two dimensions. SIAM J. Comput. 9, 846–855 (1980)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baker, B.S., Schwarz, J.S.: Shelf algorithms for two-dimensional packing problems. SIAM J. Comput. 12, 508–525 (1983)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Coffman, E.G., Garey, M.R., Johnson, D.S., Tarjan, R.E.: Performance bounds for level-oriented two-dimensional packing algorithms. SIAM J. Comput. 9, 808–826 (1980)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Csirik, J., Woeginger, G.J.: Shelf algorithms for online strip packing. Inf. Proc. Lett. 63, 171–175 (1997)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1+ ɛ in linear time. Combinatorica 1, 349–355 (1981)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Fernandez de la Vega, W., Zissimopoulos, V.: An approximation scheme for strip packing of rectangles with bounded dimensions. Discrete Appl. Math. 82, 93–101 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979Google Scholar
  9. 9.
    Gilmore, P.C., Gomory, R.E.: Multistage cutting stock problems of two and more dimensions. Oper. Res. 13, 94–120 (1965)zbMATHCrossRefGoogle Scholar
  10. 10.
    Golan, I.: Performance bounds for orthogonal, oriented two-dimensional packing algorithms. SIAM J. Comput. 10, 571–582 (1981)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin packing problem. In: Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science (FOCS’1982), 1982, pp. 312–320Google Scholar
  12. 12.
    Kenyon, C., Remila, E.: Approximate strip packing. In: Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’1996), 1996, pp. 31–36Google Scholar
  13. 13.
    Kenyon, C., Remila, E.: A near-optimal solution to a two-dimensional cutting stock problem. Math. Oper. Res. 25, 645–656 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Salzer, H.E.: The approximation of numbers as sums of reciprocals. Am. Math. Monthly 54, 135–142 (1947)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sleator, D.D.K.D.B.: A 2.5 times optimal algorithm for packing in two dimensions. Inf. Proc. Lett. 10, 37–40 (1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Steven S. Seiden
    • 1
  • Gerhard J. Woeginger
    • 2
  1. 1.Department of Computing ScienceLousiana State UniversityBaton RougeUSA
  2. 2.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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