SIAM Journal on Optimization, Volume 31, Issue 1, Page 200-216, January 2021.
We obtain a transference bound for vertices of corner polyhedra that connects two well-established areas of research: proximity and sparsity of solutions to integer programs. In the knapsack scenario, it implies that for any vertex ${x}^*$ of an integer feasible knapsack polytope ${P}({a}, { b})=\{{x} \in {\mathbb R}^n_{\ge 0}: {a}^\top{x}={ b}\}$, ${a}\in {\mathbb Z}^n_{>0}$, there exists an integer point ${z}^*\in {P}({a}, { b})$ such that, denoting by $s$ the size of the support of ${z}^*$ and assuming $s>0$, $ \|{x}^{*}-{z}^{*}\|_{\infty} \,{2^{s-1}}/{s} < \|{a}\|_{\infty}\,, $ where $\|\cdot\|_{\infty}$ stands for the $\ell_{\infty}$-norm. The bound gives an exponential in $s$ improvement on previously known proximity estimates. In addition, for general integer linear programs we obtain a resembling result that connects the minimum absolute nonzero entry of an optimal solution with the size of its support.

中文翻译：

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角多面体顶点的距离稀疏传递

SIAM优化杂志，第31卷，第1期，第200-216页，2021年1月。
我们获得了转角多面体顶点的转移边界，该转移边界连接了两个公认的研究领域：整数程序解的接近性和稀疏性。在背包场景中，这意味着对于整数可行背包多面体$ {P}（{a}，{b}）= \ {{x} \ in {\ mathbb R}中的任何顶点$ {x} ^ * $ ^ n _ {\ ge 0}：{a} ^ \ top {x} = {b} \} $，$ {a} \在{\ mathbb Z} ^ n _ {> 0} $中，存在一个整数点$ {P}（{a}，{b}）$中的{z} ^ * \，这样，用$ s $表示$ {z} ^ * $的支撑大小，并假设$ s> 0 $， \ | {x} ^ {*}-{z} ^ {*} \ | _ {\ infty} \，{2 ^ {s-1}} / {s} <\ | {a} \ | _ {\ infty} \ ,, $，其中$ \ | \ cdot \ | __ {\ infty} $代表$ \ ell _ {\ infty} $范数。该界限在先前已知的邻近度估计值上给出了$ s $的指数改进。此外，