Abstract
In the min-knapsack problem one aims at choosing a set of objects with minimum total cost and total profit above a given threshold. In this paper, we study a class of valid inequalities for min-knapsack known as bounded pitch inequalities, which generalize the well-known unweighted cover inequalities. While separating over pitch-1 inequalities is NP-Hard, we show that approximate separation over the set of pitch-1 and pitch-2 inequalities can be done in polynomial time. We also investigate integrality gaps of linear relaxations for min-knapsack when these inequalities are added. Among other results, we show that, for any fixed t, the t-th CG closure of the natural linear relaxation has the unbounded integrality gap.
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Notes
- 1.
Note that \(c \in \mathbb {R}_+^n, \ p \in \mathbb {R}_+^n\) and the constraint is scaled so that the right-hand side is 1.
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Acknowledgments
Supported by the Swiss National Science Foundation (SNSF) project 200020-169022 “Lift and Project Methods for Machine Scheduling Through Theory and Experiments”. Some of the work was done when the second and the third author visited the IEOR department of Columbia University, partially funded by a gift of the SNSF.
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Faenza, Y., Malinović, I., Mastrolilli, M., Svensson, O. (2018). On Bounded Pitch Inequalities for the Min-Knapsack Polytope. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_15
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