In the three-dimensional (3D) strip packing problem, we are given a set of 3D rectangular items and a 3D box $B$. The goal is to pack all the items in $B$ such that the height of the packing is minimized. We consider the most basic version of the problem, where the items must be packed with their edges parallel to the edges of $B$ and cannot be rotated. Building upon Caprara's work for the two-dimensional (2D) bin packing problem, we obtain an algorithm that, given any $\epsilon>0$, achieves an approximation of $T_{\infty}+\epsilon\approx1.69103+\epsilon$, where $T_{\infty}$ is the well-known number that occurs naturally in the context of bin packing. Our key idea is to establish a connection between bin packing solutions for an arbitrary instance $I$ and the strip packing solutions for the corresponding instance obtained from $I$ by applying the harmonic transformation to certain dimensions. Based on this connection, we also give a simple alternate proof of the $T_{\infty}+\epsilon$ approximation for 2D bin packing due to Caprara. In particular, we show how his result follows from a simple modification of the asymptotic approximation scheme for 2D strip packing due to Kenyon and Rémila.

  • 1.  B. S. Baker E. G. Coffman and  R. L. Rivest , Orthogonal packings in two dimensions , SIAM J. Comput. , 9 ( 1980 ), pp. 846 -- 855 . LinkISIGoogle Scholar

  • 2.  N. Bansal A. Caprara and  M. Sviridenko , Improved approximation algorithms for multidimensional bin packing problems , in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science , 2006 , pp. 697 -- 708 . Google Scholar

  • 3.  N. Bansal A. Caprara and  M. Sviridenko , A new approximation method for set covering problems, with applications to multidimensional bin packing , SIAM J. Comput. , 39 ( 2009 ), pp. 1256 -- 1278 . LinkISIGoogle Scholar

  • 4.  N. Bansal J. Correa C. Kenyon and  M. Sviridenko , Bin packing in multiple dimensions: Inapproximability results and approximation schemes , Math. Oper. Res. , 31 ( 2006 ), pp. 31 -- 49 . CrossrefISIGoogle Scholar

  • 5.  A. Caprara , Packing two-dimensional bins in harmony , in Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science , 2002 , pp. 490 -- 499 . Google Scholar

  • 6.  A. Caprara , Packing d-dimensional bins in d stages , Math. Oper. Res. , 33 ( 2008 ), pp. 203 -- 215 . CrossrefISIGoogle Scholar

  • 7.  E. G. Coffman M. R. Garey D. S. Johnson and  R. E. Tarjan , Performance bounds for level-oriented two dimensional packing algorithms , SIAM J. Comput. , 9 ( 1980 ), pp. 808 -- 826 . LinkISIGoogle Scholar

  • 8.  J. R. Correa , Resource augmentation in two-dimensional packing with orthogonal rotations , Oper. Res. Lett. , 34 ( 2006 ), pp. 85 -- 93 . CrossrefISIGoogle Scholar

  • 9.  W. Fernandez de la Vega and  G. S. Lueker , Bin packing can be solved within 1+ epsilon in linear time , Combinatorica , 1 ( 1981 ), pp. 349 -- 355 . CrossrefISIGoogle Scholar

  • 10.  X. Han K. Iwama and  G. Zhang , Strip packing vs. bin packing , in Proceedings of the 3rd International Conference on Algorithmic Aspects in Informatiom and Management, Springer-Verlag , Berlin , 2007 , pp. 358 -- 367 . Google Scholar

  • 11.  D. S. Johnson , Near-Optimal Bin Packing Algorithms , Doctoral thesis, MIT , Cambridge, MA , 1973 . Google Scholar

  • 12.  K. Jansen and  R. Solis-Oba , An asymptotic approximation algorithm for 3D-strip packing , in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms , 2006 , pp. 143 -- 152 . Google Scholar

  • 13.  N. Karmarkar and  R. M. Karp , An efficient approximation scheme for the one-dimensional bin-packing problem , in Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science , 1982 , pp. 312 -- 320 . Google Scholar

  • 14.  C. Kenyon and  E. Rémila , A near-optimal solution to a two-dimensional cutting stock problem , Math. Oper. Res. , 25 ( 2000 ), pp. 645 -- 656 . CrossrefISIGoogle Scholar

  • 15.  K. Li and  K.-H. Cheng , On three-dimensional packing , SIAM J. Comput. , 19 ( 1990 ), pp. 847 -- 867 . LinkISIGoogle Scholar

  • 16.  K. Li . Cheng, Heuristic algorithms for on-line packing in three dimensions , J. Algorithms , 13 ( 1992 ), pp. 589 -- 605 . CrossrefISIGoogle Scholar

  • 17.  C. Lee and  D. Lee , A simple on-line bin-packing algorithm , J. ACM , 32 ( 1985 ), pp. 562 -- 572 . CrossrefISIGoogle Scholar

  • 18.  A. Meir and  L. Moser , On packing of squares and cubes , J. Combin. Theory , 5 ( 1968 ), pp. 126 -- 134 . CrossrefGoogle Scholar

  • 19.  F. Miyazawa and  Y. Wakabayashi , An algorithm for the three-dimensional packing problem with asymptotic performance analysis , Algorithmica , 18 ( 1997 ), pp. 122 -- 144 . CrossrefISIGoogle Scholar

  • 20.  S. A. Plotkin D. B. Shmoys and  E. Tardos , Fast approximation algorithms for fractional packing and covering problems , Math. Oper. Res. , 20 ( 1995 ), pp. 257 -- 301 . CrossrefISIGoogle Scholar

  • 21.  N. J. A. Sloane, Sequences A000058 and A007018, The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A000058 and http://oeis.org/A007018.Google Scholar