Two for One: Tight Approximation of 2D Bin Packing

In this paper, we study the two-dimensional geometrical bin packing problem (2DBP): given a list of rectangles, provide a packing of all these into the smallest possible number of 1×1 bins without rotating the rectangles. We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the rotational case and also matches the known lower bound of approximability. Our approach makes strong use of a recently-discovered PTAS for a related knapsack problem and a new algorithm that can pack instances into OPT + 2 bins for any constant OPT.

[1]  Klaus Jansen,et al.  Maximizing the Total Profit of Rectangles Packed into a Rectangle , 2007, Algorithmica.

[2]  Rob van Stee,et al.  Packing Rectangles into 2OPT Bins Using Rotations , 2008, SWAT.

[3]  Alberto Caprara,et al.  A New Approximation Method for Set Covering Problems, with Applications to Multidimensional Bin Packing , 2009, SIAM J. Comput..

[4]  Alberto Caprara,et al.  Packing d-Dimensional Bins in d Stages , 2008, Math. Oper. Res..

[5]  Klaus Jansen,et al.  Rectangle packing with one-dimensional resource augmentation , 2009, Discret. Optim..

[6]  Nikhil Bansal,et al.  New approximability and inapproximability results for 2-dimensional Bin Packing , 2004, SODA '04.

[7]  Klaus Jansen,et al.  A Structural Lemma in 2-Dimensional Packing, and Its Implications on Approximability , 2009, ISAAC.

[8]  Brenda S. Baker,et al.  A 5/4 Algorithm for Two-Dimensional Packing , 1981, J. Algorithms.

[9]  Alberto Caprara,et al.  Packing 2-dimensional bins in harmony , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[10]  A. Steinberg,et al.  A Strip-Packing Algorithm with Absolute Performance Bound 2 , 1997, SIAM J. Comput..

[11]  Klaus Jansen,et al.  New Approximability Results for 2-Dimensional Packing Problems , 2007, MFCS.

[12]  Alberto Caprara,et al.  Improved approximation algorithms for multidimensional bin packing problems , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[13]  Ronald L. Rivest,et al.  Orthogonal Packings in Two Dimensions , 1980, SIAM J. Comput..

[14]  Joseph Naor,et al.  Approximating the Advertisement Placement Problem , 2002, J. Sched..

[15]  Wolfgang Maass,et al.  Approximation Schemes for Covering and Packing Problems in Robotics and VLSI , 1984, STACS.

[16]  Guochuan Zhang,et al.  A 3-approximation algorithm for two-dimensional bin packing , 2005, Oper. Res. Lett..

[17]  Joseph Y.-T. Leung,et al.  Packing Squares into a Square , 1990, J. Parallel Distributed Comput..

[18]  Klaus Jansen,et al.  On rectangle packing: maximizing benefits , 2004, SODA '04.

[19]  Ronald L. Graham,et al.  Bounds on Multiprocessing Timing Anomalies , 1969, SIAM Journal of Applied Mathematics.

[20]  Klaus Jansen,et al.  A Polynomial Time Approximation Scheme for the Square Packing Problem , 2008, IPCO.

[21]  Claire Mathieu,et al.  A Near-Optimal Solution to a Two-Dimensional Cutting Stock Problem , 2000, Math. Oper. Res..