Packing Groups of Items into Multiple Knapsacks

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 which are 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 constantapproximation 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.

Bi-Criteria Multiple Knapsack Problem with Grouped Items

“…This paper studies GMKP, a strongly N P-hard problem with no polynomial time approximation algorithm. We are not the first studying GMKP [Chen and Zhang, 2018], but we are the first studying the bi-criteria version of GMKP and offering a broad computational study of our suggested algorithms and heuristics. We propose several pseudo-polynomial time approximation algorithms for bi-GMKP with tight guarantees, that can be adapted as binary-search heuristics for GMKP, and heuristics for bi-GMKP.…”

“…There is a PTAS, but not an FPTAS for MKP; even for two knapsacks [Chekuri and Khanna, 2005]. There is no PTAS nor constant-ratio approximation algorithm for GMKP [Chen and Zhang, 2018].…”

“…For a special case of GMKP, when all knapsack capacities are equal and the heaviest group weighs at most 2/3 of the total capacity, parameterized-approximation algorithms were proposed by Chen and Zhang [2018]. Adany et al [2016] proposed a PTAS for a generalized assignment problem with grouped items that has additional constraints: there is a limit on the number of items per group, and knapsacks can accommodate at most one item from each group.…”

“…Adany et al [2016] proposed a PTAS for a generalized assignment problem with grouped items that has additional constraints: there is a limit on the number of items per group, and knapsacks can accommodate at most one item from each group. The algorithms proposed by Chen and Zhang [2018] and Adany et al [2016] guarantee feasiblity while sacrificing rewards; the algorithms proposed in this paper sacrifice feasibility (bounded by a maximum exceeded knapsack capacity) while generating solutions achieving the optimal reward (in relation to the original GMKP).…”

Generalized Assignment via Submodular Optimization with Reserved Capacity

“…Chen and Zhang [11] studied the problem of group packing of items into multiple knapsacks (GMKP), a special case of Group GAP where the profit of each item is the same across the XX:2 Generalized Assignment via Submodular Optimization bins. Let GMKP(δ) be the restriction of GMPK to instances in which the total size of items in each group is at most δm (that is, a factor δ of the total capacity of all bins).…”