I am currently trying to figure out whether a flavor of the bin packing problem, which I call the "flexible bin packing problem" (F-BPP), is NP-complete.
Here are the definitions for the traditional bin packing problem (BPP) and the flexible bin packing problem (F-BPP):
Bin packing problem (BPP):
Optimization problem: Given a set of n items and their corresponding volumes (the volume of each item is between 0 and 1), pack the n items into the minimum number of bins (each with capacity 1) such that the capacity of each bin is not exceeded (https://ics.uci.edu/~goodrich/teach/cs165/notes/BinPacking.pdf).
Decision problem: Given a set of n items and their corresponding volumes (the volume of each item is between 0 and 1), and an integer k, determine if it is possible to pack all n items into at most k bins, each with capacity 1, such that the capacity of each bin is not exceeded. This problem is NP-complete (https://ics.uci.edu/~goodrich/teach/cs165/notes/BinPacking.pdf).
Flexible bin packing problem (F-BPP):
Decision problem: Given n item types, m bin types, and a mapping between compatible item type and bin type pairs (the pairs are keys in the mapping) and the volume (between 0 and 1) of an item of that type in a bin of that type (the volumes are values in mapping), and an integer k >= m, determine if it is possible to pack at least one item of each type in at most k bins, where each bin has capacity 1, the capacity of each bin is not exceeded, and all bin types are used at least once. Items and bin types can be used more than once.