Abstract
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.
Highlights
The classical multiple knapsack problem aims at a most profitable subset of given items which admits a feasible packing on a given set of knapsacks
groups of items into multiple knapsacks (GMKP) does not admit any constant ratio approximation algorithm as it is easy to see that deciding whether a single group of items could be packed into m = 2 bins is exactly the Partition problem and is NP-complete
We first provide a brief overview on the classical multiple knapsack problem (MKP)
Summary
The classical multiple knapsack problem aims at a most profitable subset of given items which admits a feasible packing on a given set of knapsacks. Most of the times it is reasonable to assume that every single item has an individual profit, as we do in the classical (multiple) knapsack problem, it does happen in many cases that the profit can only be defined for a group of items, instead of each of them. GMKP does not admit any constant ratio approximation algorithm as it is easy to see that deciding whether a single group of items (with the profit of 1) could be packed into m = 2 bins is exactly the Partition problem and is NP-complete. The intractability of the problem follows from the fact that a single group may have a weight as large as the total capacity of all the knapsacks (bins), which is often not the case in practice. For the sake of conventional convenience, if the factor c is arbitrarily small, we say such an algorithm does not have a constant ratio
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