Abstract
Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, for example, in scheduling and stock-cutting, and has been studied extensively.
When the dimensions of the objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than 3/2, unless P= NP. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a (1.4 + ϵ)-approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values are allowed in the input. Their result has subsequently been improved to a (4/3 + ϵ)-approximation by two independent research groups [FSTTCS 2016, WALCOM 2017]. This raises a question whether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related two-dimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack.
In this article, we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than 12/11, even when restricted to polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP} ⊑ DTIME(2polylog (n)).
- Anna Adamaszek and Andreas Wiese. 2013. Approximation schemes for maximum weight independent set of rectangles. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS’13), Omer Reingold (Ed.). IEEE Computer Society, 400--409. Google Scholar
Digital Library
- Anna Adamaszek and Andreas Wiese. 2015. A quasi-PTAS for the two-dimensional geometric knapsack problem. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15), Piotr Indyk (Ed.). SIAM, 1491--1505. Google Scholar
Digital Library
- Brenda S. Baker, Donna J. Brown, and Howard P. Katseff. 1981. A 5/4 algorithm for two-dimensional packing. J. Algor. 2, 4 (1981), 348--368.Google Scholar
Cross Ref
- Brenda S. Baker, Edward G. Coffman Jr., and Ronald L. Rivest. 1980. Orthogonal packings in two dimensions. SIAM J. Comput. 9, 4 (1980), 846--855.Google Scholar
Digital Library
- Waldo Gálvez, Fabrizio Grandoni, Salvatore Ingala, and Arindam Khan. 2016. Improved pseudo-polynomial-time approximation for strip packing. In Proceedings of the 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’16) (LIPIcs), Akash Lal, S. Akshay, Saket Saurabh, and Sandeep Sen (Eds.), Vol. 65. Schloss Dagstuhl--Leibniz-Zentrum für Informatik, 9:1--9:14.Google Scholar
- Michael R. Garey and David S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman 8 Co., New York, NY. Google Scholar
Digital Library
- Igal Golan. 1981. Performance bounds for orthogonal oriented two-dimensional packing algorithms. SIAM J. Comput. 10, 3 (1981), 571--582.Google Scholar
Digital Library
- Rolf Harren, Klaus Jansen, Lars Prädel, and Rob van Stee. 2014. A (5/3 + )-approximation for strip packing. Computat. Geom. 47, 2 (2014), 248--267. Google Scholar
Digital Library
- Rolf Harren and Rob van Stee. 2009. Improved absolute approximation ratios for two-dimensional packing problems. In Proceedings of the 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’09), and the 13th International Workshop on Randomization and Computation (RANDOM’09) (LNCS), Irit Dinur, Klaus Jansen, Joseph Naor, and José D. P. Rolim (Eds.), Vol. 5687. Springer, 177--189. Google Scholar
Digital Library
- Klaus Jansen and Malin Rau. 2017. Improved approximation for two dimensional strip packing with polynomial bounded width. In Proceedings of the 11th International Conference and Workshops on Algorithms and Computation (WALCOM’17) (LNCS), Sheung-Hung Poon, Md. Saidur Rahman, and Hsu-Chun Yen (Eds.), Vol. 10167. Springer, 409--420.Google Scholar
Cross Ref
- Klaus Jansen and Roberto Solis-Oba. 2009. Rectangle packing with one-dimensional resource augmentation. Discrete Optim. 6, 3 (2009), 310--323. Google Scholar
Digital Library
- Klaus Jansen and Rob van Stee. 2005. On strip packing with rotations. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC’05), Harold N. Gabow and Ronald Fagin (Eds.). ACM, 755--761. Google Scholar
Digital Library
- Klaus Jansen and Guochuan Zhang. 2007. Maximizing the total profit of rectangles packed into a rectangle. Algorithmica 47, 3 (2007), 323--342.Google Scholar
Digital Library
- Edward G. Coffman Jr., M. R. Garey, David S. Johnson, and Robert Endre Tarjan. 1980. Performance bounds for level-oriented two-dimensional packing algorithms. SIAM J. Comput. 9, 4 (1980), 808--826.Google Scholar
Digital Library
- Claire Kenyon and Eric Rémila. 2000. A near-optimal solution to a two-dimensional cutting stock problem. Math. Oper. Res. 25, 4 (2000), 645--656. Google Scholar
Digital Library
- Giorgi Nadiradze and Andreas Wiese. 2016. On approximating strip packing with a better ratio than 3/2. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’16), Robert Krauthgamer (Ed.). SIAM, 1491--1510. Google Scholar
Digital Library
- Ingo Schiermeyer. 1994. Reverse-Fit: A 2-optimal algorithm for packing rectangles. In Proceedings of the 2nd Annual European Symposium on Algorithms, ESA 1994 (LNCS’94), Jan van Leeuwen (Ed.), Vol. 855. Springer, 290--299. Google Scholar
Digital Library
- Daniel Dominic Sleator. 1980. A 2.5 times optimal algorithm for packing in two dimensions. Inform. Process. Lett. 10, 1 (1980), 37--40.Google Scholar
Cross Ref
- A. Steinberg. 1997. A strip-packing algorithm with absolute performance bound 2. SIAM J. Comput. 26, 2 (1997), 401--409. Google Scholar
Digital Library
Index Terms
Hardness of Approximation for Strip Packing
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