Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes

  • Authors:

  • Nikhil Bansal;Jos R. Correa;Claire Kenyon;Maxim Sviridenko

  • Affiliations:

  • IBM T. J. Watson Research Center, Yorktown Heights, New York 10598;School of Business, Univesidad Adolfo Ibez, Avenida Presidente Errzuriz 3485, Las Condes, Santiago, Chile;Department of Computer Science, Brown University, Providence, Rhode Island 02912;IBM T. J. Watson Research Center, Yorktown Heights, New York 10598

  • Venue:
  • Mathematics of Operations Research
  • Year:
  • 2006

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Abstract

We study the following packing problem: Given a collection of d-dimensional rectangles of specified sizes, pack them into the minimum number of unit cubes. We show that unlike the one-dimensional case, the two-dimensional packing problem cannot have an asymptotic polynomial time approximation scheme (APTAS), unless PNP. On the positive side, we give an APTAS for the special case of packing d-dimensional cubes into the minimum number of unit cubes. Second, we give a polynomial time algorithm for packing arbitrary two-dimensional rectangles into at most OPT square bins with sides of length 1 , where OPT denotes the minimum number of unit bins required to pack these rectangles. Interestingly, this result has no additive constant term, i.e., is not an asymptotic result. As a corollary, we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum-area encasing rectangle.