TY - JOUR

T1 - Packing groups of items into multiple knapsacks

AU - Chen, Lin

AU - Zhang, Guochuan

N1 - Publisher Copyright:
© 2018 ACM.

PY - 2018/8

Y1 - 2018/8

N2 - 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.

AB - 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.

KW - Approximation schemes

KW - Bin packing

KW - Lower bounds

KW - Multiple knapsack

UR - http://www.scopus.com/inward/record.url?scp=85052602353&partnerID=8YFLogxK

U2 - 10.1145/3233524

DO - 10.1145/3233524

M3 - Article

AN - SCOPUS:85052602353

VL - 14

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 4

M1 - 51

ER -