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 NPhard 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 delta of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to delta 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 delta, changes substantially, which has a clear difference from the classical multiple knapsack problem.
BibTeX  Entry
@InProceedings{chen_et_al:LIPIcs:2016:5729,
author = {Lin Chen and Guochuan Zhang},
title = {{Packing Groups of Items into Multiple Knapsacks}},
booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
pages = {28:128:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770019},
ISSN = {18688969},
year = {2016},
volume = {47},
editor = {Nicolas Ollinger and Heribert Vollmer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5729},
URN = {urn:nbn:de:0030drops57299},
doi = {10.4230/LIPIcs.STACS.2016.28},
annote = {Keywords: approximation algorithms, lower bound, multiple knapsack, bin packing}
}
Keywords: 

approximation algorithms, lower bound, multiple knapsack, bin packing 
Collection: 

33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) 
Issue Date: 

2016 
Date of publication: 

16.02.2016 