Packing groups of items into multiple knapsacks

Lin Chen, Guochuan Zhang

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


We consider a natural generalization of the classical multiple knapsack problem in which instead of packing single items we are packing groups of items. In this problem, we have multiple knapsacks and a set of items partitioned into groups. Each item has an individual weight, while the profit is associated with groups rather than items. The profit of a group can be attained if and only if every item of this group is packed. Such a general model finds applications in various practical problems, e.g., delivering bundles of goods. The tractability of this problem relies heavily on how large a group could be. Deciding if a group of items of total weight 2 could be packed into two knapsacks of unit capacity is already NP-hard and it thus rules out a constant-approximation algorithm for this problem in general. We then focus on the parameterized version where the total weight of items in each group is bounded by a factor δ of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to δ are derived. We also show that, depending on whether the number of knapsacks is a constant or part of the input, the approximation ratio for the problem, as a function on δ, changes substantially, which has a clear difference from the classical multiple knapsack problem.

Original languageEnglish
Article number51
JournalACM Transactions on Algorithms
Issue number4
StatePublished - Aug 2018


  • Approximation schemes
  • Bin packing
  • Lower bounds
  • Multiple knapsack


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