Distances to lattice points in knapsack polyhedra

We give an optimal upper bound for the ℓ ∞ -distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integra...

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Bibliographic Details
Published in:Mathematical programming 2020-07, Vol.182 (1-2), p.175-198
Main Authors: Aliev, Iskander, Henk, Martin, Oertel, Timm
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Full Length Paper
Integer programming
Integers
Knapsack problem
Lower bounds
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Polyhedra
Production scheduling
Theoretical
Upper bounds
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Summary:We give an optimal upper bound for the ℓ ∞ -distance from a vertex of a knapsack polyhedron to its nearest feasible lattice point. In a randomised setting, we show that the upper bound can be significantly improved on average. As a corollary, we obtain an optimal upper bound for the additive integrality gap of integer knapsack problems and show that the integrality gap of a “typical” knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario. We also prove that, in a generic case, the integer programming gap admits a natural optimal lower bound.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01392-1