scholarly journals A tight lower bound for the online bounded space hypercube bin packing problem

2021 ◽  
Vol vol. 23, no. 3 (Discrete Algorithms) ◽  
Author(s):  
Yoshiharu Kohayakawa ◽  
Flávio Keidi Miyazawa ◽  
Yoshiko Wakabayashi
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Performance Ratio ◽  
Asymptotic Performance ◽  
Bin Packing Problem ◽  
Bounded Space ◽  
Space Variant

In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $\rho$ of the online bounded space variant is $\Omega(\log d)$ and $O(d/\log d)$, and conjectured that it is $\Theta(\log d)$. We show that $\rho$ is in fact $\Theta(d/\log d)$, using probabilistic arguments.

A new lower bound for the non-oriented two-dimensional bin-packing problem

2007 ◽  
Vol 35 (3) ◽  
pp. 365-373 ◽  
Author(s):  
François Clautiaux ◽  
Antoine Jouglet ◽  
Joseph El Hayek
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Two Dimensional ◽  
Bin Packing Problem

TWO FOR ONE: TIGHT APPROXIMATION OF 2D BIN PACKING

2013 ◽  
Vol 24 (08) ◽  
pp. 1299-1327 ◽  
Author(s):  
ROLF HARREN ◽  
KLAUS JANSEN ◽  
LARS PRÄDEL ◽  
ULRICH M. SCHWARZ ◽  
ROB VAN STEE
Keyword(s):  
Lower Bound ◽  
Knapsack Problem ◽  
Bin Packing ◽  
Packing Problem ◽  
Approximate Algorithm ◽  
Two Dimensional ◽  
Chip Design ◽  
Practical Applications ◽  
Vlsi Chip ◽  
Bin Packing Problem

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 unit bins without rotating the rectangles. Beyond its theoretical appeal, this problem has many practical applications, for example in print layout and VLSI chip design. We present a 2-approximate algorithm, which improves over the previous best known ratio of 3, matches the best results for the problem where rotations are allowed and also matches the known lower bound of approximability. Our approach makes strong use of a PTAS for a related 2D knapsack problem and a new algorithm that can pack instances into two bins if OPT = 1.

Lower Bound for the Online Bin Packing Problem with Restricted Repacking

2008 ◽  
Vol 38 (1) ◽  
pp. 398-410 ◽  
Author(s):  
János Balogh ◽  
József Békési ◽  
Gábor Galambos ◽  
Gerhard Reinelt
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Bin Packing Problem

scholarly journals A lower bound for the non-oriented two-dimensional bin packing problem

2002 ◽  
Vol 118 (1-2) ◽  
pp. 13-24 ◽  
Author(s):  
Mauro Dell'Amico ◽  
Silvano Martello ◽  
Daniele Vigo
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Two Dimensional ◽  
Bin Packing Problem

scholarly journals An improved lower bound for the bin packing problem

1996 ◽  
Vol 66 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Bintong Chen ◽  
Bharatendu Srivastava
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Bin Packing Problem

An analysis of lower bound procedures for the bin packing problem

2005 ◽  
Vol 32 (3) ◽  
pp. 395-405 ◽  
Author(s):  
Jean-Marie Bourjolly ◽  
Vianney Rebetez
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Bin Packing Problem

Chance-Constrained Multiple Bin Packing Problem with an Application to Operating Room Planning

2021 ◽  
Author(s):  
Shanshan Wang ◽  
Jinlin Li ◽  
Sanjay Mehrotra
Keyword(s):  
Lower Bound ◽  
Operating Room ◽  
Operations Research ◽  
Bin Packing ◽  
Packing Problem ◽  
Real Data ◽  
Branch And Cut ◽  
Computational Experiments ◽  
Operating Room Planning ◽  
Bin Packing Problem

We study the chance-constrained bin packing problem, with an application to hospital operating room planning. The bin packing problem allocates items of random sizes that follow a discrete distribution to a set of bins with limited capacity, while minimizing the total cost. The bin capacity constraints are satisfied with a given probability. We investigate a big-M and a 0-1 bilinear formulation of this problem. We analyze the bilinear structure of the formulation and use the lifting techniques to identify cover, clique, and projection inequalities to strengthen the formulation. We show that in certain cases these inequalities are facet-defining for a bilinear knapsack constraint that arises in the reformulation. An extensive computational study is conducted for the operating room planning problem that minimizes the number of open operating rooms. The computational tests are performed using problems generated based on real data from a hospital. A lower-bound improvement heuristic is combined with the cuts proposed in this paper in a branch-and-cut framework. The computations illustrate that the techniques developed in this paper can significantly improve the performance of the branch-and-cut method. Problems with up to 1,000 scenarios are solved to optimality in less than an hour. A safe approximation based on conditional value at risk (CVaR) is also solved. The computations show that the CVaR approximation typically leaves a gap of one operating room (e.g., six instead of five) to satisfy the chance constraint. Summary of Contribution: This paper investigates a branch-and-cut algorithm for a chance-constrained bin packing problem with multiple bins. The chance-constrained bin packing provides a modeling framework for applied operations research problems, such as health care, scheduling, and so on. This paper studies alternative computational approaches to solve this problem. Moreover, this paper uses real data from a hospital operating room planning setting as an application to test the algorithmic ideas. This work, therefore, is at the intersection of computing and operations research. Several interesting ideas are developed and studied. These include a strengthened big-M reformulation, analysis of a bilinear reformulation, and identifying certain facet-defining inequalities for this formulation. This paper also gives a lower-bound generation heuristic for a model that minimizes the number of bins. Computational experiments for an operating room planning model that uses data from a hospital demonstrate the computational improvement and importance of the proposed approaches. The techniques proposed in this paper and computational experiments further enhance the interface of computing and operations research.

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