An Improved Knapsack Solver for Column Generation
The Knapsack Problem (KP) and its variants are well-known NP-hard problems. Their study is also driven by approximation algorithms for optimization problems like Bin Packing: these algorithms must often solve KP instances as subproblems. In this paper, we introduce the Knapsack Problem with Inversely Proportional Profits (KPIP), a generalization of KP: in it, one of several knapsack sizes has to be chosen. At the same time, the item profits are inversely proportional to the chosen knapsack size so that it is non-trivial to take the right knapsack. We adapt Lawler’s approximation scheme for KP to faster solve KPIP. Thus, we are able to improve the running time of an approximation scheme for Variable-Sized Bin Packing that solves KPIP as a subproblem.
KeywordsKnapsack Problem Unbounded Knapsack Problem Bounded Knapsack Problem Variable-Sized Bin Packing Column Generation
Unable to display preview. Download preview PDF.
- 1.Bellman, R.E.: Dynamic Programming. Princeton University Press (1957)Google Scholar
- 2.Garey, M., Johnson, D.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman and Company (1979)Google Scholar
- 7.Jansen, K., Kraft, S.: An improved approximation scheme for variable-sized bin packing. Tech. Rep. 1301, Christian-Albrechts-Universität zu Kiel (2013) ISSN 2192-6247Google Scholar