Distance-Sparsity Transference for Lattice Points in Knapsack Polytopes

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Proves a quantitative transference: a concrete inequality that converts a distance-to-vertex bound for a knapsack polytope into an explicit sparsity bound on integer solutions, with a removed nuisance parameter and provable improvement (examples show up to a threefold factor improvement in the regime studied). Tools are geometry-of-numbers projections and lattice constructions; the argument produces explicit integer vectors witnessing the bound and explains which knapsack data give the strongest gains.

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" chapters (9)

1. 0:00Intro
2. 1:13Two research directions in the theory of Integer Programming (IP)
3. 3:47Distance (proximity) bounds: knapsack scenario
4. 10:17Distance bounds for general IP polyhedra
5. 11:31Sparsity bounds: knapsack scenario
6. 12:49Sparsity bounds for general IP polyhedra
7. 13:53Distance/sparsity transference
8. 19:39Idea of the proof of Theorem 5
9. 22:24Transference for general IP polyhedra?