Bin packing problem
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| Bin packing problem | |
|---|---|
| Name | Bin packing problem |
| Input | Multiset of item sizes, bin capacity |
| Output | Partition of items into bins |
| Goal | Minimize number of bins used |
| Complexity | NP-hard, strongly NP-hard |
| Applications | Logistics, scheduling, memory allocation |
Bin packing problem The bin packing problem asks how to pack a multiset of items of given sizes into a minimum number of fixed-capacity bins. Originating in combinatorial optimization and operations research, the problem connects to practical tasks in FedEx logistics, Intel memory management, Fedora package distribution, and industrial cutting-stock contexts. Research spans exact algorithms, approximation schemes, and heuristics applied in contexts such as Amazon warehousing, Boeing production planning, and high-performance computing at institutions like Los Alamos National Laboratory.
Definition and variants
The canonical formulation specifies a bin capacity C and items with positive sizes s_i; the objective is to partition items into bins so each bin's total size ≤ C while minimizing bin count. Variants include the one-dimensional, two-dimensional, and three-dimensional packing used in IKEA logistics and IKEA-style furniture shipping; the online variant where items arrive sequentially in settings like UPS parcel routing; the variable-sized bin variant relevant to United States Postal Service rate-based container choice; the bin packing with cardinality constraints used in Intel processor register allocation; and the cutting-stock variant historically studied alongside operations at General Motors and Ford Motor Company.
Complexity and computational hardness
Deciding whether items fit into k bins is NP-complete, a result with roots in reductions from Partition problem-like instances studied by early theoreticians at Bell Labs and universities such as Princeton University and Massachusetts Institute of Technology. The optimization version is strongly NP-hard, making polynomial-time exact algorithms unlikely under assumptions related to P versus NP problem and its implications for institutions like Clay Mathematics Institute. Hardness also manifests in online adversarial models considered in work connected to Microsoft Research and lower bounds proved by complexity theorists at University of California, Berkeley and University of Cambridge.
Exact algorithms and optimal solutions
Exact methods include integer linear programming formulations solved by solvers from companies such as IBM (CPLEX) and Gurobi; branch-and-bound and branch-and-price techniques developed in research groups at ETH Zurich and INRIA; dynamic programming approaches used in theoretical work at Stanford University; and cutting-plane methods exploited in industrial applications at Siemens. For small-instance or fixed-parameter regimes, algorithms leveraging bounded item sizes or fixed bin counts have been implemented in laboratories at Carnegie Mellon University and Cornell University to produce provably optimal packings.
Approximation algorithms and heuristics
Approximation schemes include asymptotic polynomial-time approximation schemes (APTAS) and fully polynomial-time approximation schemes (FPTAS) developed in collaborations including researchers at University of Oxford and Technische Universität Berlin. Classical heuristics such as First-Fit, Best-Fit, First-Fit Decreasing, and Best-Fit Decreasing—analyzed in seminars at Princeton University and University of Washington—are widely used in industry at DHL and FedEx. Harmonic algorithms, shelf and guillotine heuristics for higher-dimensional variants, and metaheuristics (genetic algorithms, simulated annealing) have been applied in projects with Boeing and Airbus for cargo loading optimization.
Special cases and practical applications
Special-case tractable variants include bin capacity normalization leading to pseudo-polynomial dynamic programs used in embedded systems at ARM Holdings and memory packing in microcontrollers at Texas Instruments. Two-dimensional guillotine-constrained packing is used in sheet-metal manufacturing at General Electric and textile cutting at Levi Strauss & Co.. Real-time online packing algorithms support cloud providers like Google (company) and Amazon Web Services for virtual machine placement and resource binning. Supply-chain implementations have been adopted by retailers such as Walmart and Target Corporation for palletization and cartonization.
Related problems and reductions
Bin packing is closely related to partitioning and scheduling problems, including the partition problem historically studied at Bell Labs, the multiprocessor scheduling (makespan) problem researched at California Institute of Technology, and the cutting-stock problem addressed in industrial research at Tata Steel. Reductions connect it to subset sum and knapsack problems analyzed at University of Illinois Urbana–Champaign and to bin covering variants investigated in collaboration between MIT and ETH Zurich.