Multidimensional Bin Packing and Other Related Problems: A Survey

The bin packing problem is a well-studied problem in combinatorial optimization. In the classical bin packing problem, we are given a list of real numbers in (0, 1] and the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. The problem is extremely important in practice and finds numerous applications in scheduling, routing and resource allocation problems. Theoretically the problem has rich connections with discrepancy theory, iterative methods, entropy rounding and has led to the development of several algorithmic techniques. In this survey we consider several classical generalizations of bin packing problem such as geometric bin packing, vector bin packing and various other related problems. In two-dimensional geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. This variant has a lot of applications in cutting stock, vehicle loading, pallet packing, memory allocation and several other logistics and robotics related problems. In d-dimensional vector bin packing, each item is a d-dimensional vector that needs to be packed into unit vector bins. This problem is of great significance in resource constrained scheduling and in recent virtual machine placement in cloud computing. We also consider several other generalizations of bin packing such as geometric knapsack, strip packing and other related problems such as vector scheduling, vector covering etc. We survey algorithms for these problems in offline and online setting, and also mention results for several important special cases. We briefly mention related techniques used in the design and analysis of these algorithms and also survey some heuristics that work well in practice. In the end we conclude with a list of open problems. ∗This research was supported by NSF EAGER award grants CCF-1415496 and CCF-1415498 †Georgia Institute of Technology, Atlanta, USA. Email: [email protected] ‡Istituto Dalle Molle di studi sull’Intelligenza Artificiale (IDSIA), Scuola universitaria professionale della Svizzera italiana (SUPSI), Universit della Svizzera italiana (USI), Switzerland. email: [email protected]. Supported by SNF Grant 200021 159697/1. A part of this work was done when the author was a student at Georgia Institute of Technology. §Georgia Institute of Technology, Atlanta, USA. Email: [email protected] ¶Georgia Institute of Technology, Atlanta, USA. Email: [email protected]