We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form \(\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}\) where the constraint matrix \({\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}\) consists roughly of n repetitions of a matrix \(A \in {\mathbb {Z}}^{r \times s}\) on the vertical line and n repetitions of a matrix \(B \in {\mathbb {Z}}^{r \times t}\) on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and \(\Delta \), where \(\Delta \) is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time \(f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)\), where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

我们考虑所谓的 2 阶段随机整数程序 (IP) 及其广义形式，即所谓的多阶段随机 IP。一个 2 阶段随机 IP 是一个形式为\(\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \在 {\mathbb {Z}}^{s + nt} \}\)其中约束矩阵\({\mathcal {A}} \in {\mathbb {Z}}^{rn \times s +nt}\ )大致由垂直线上的矩阵\(A \in {\mathbb {Z}}^{r \times s}\)的n次重复和矩阵\ (B \in {\mathbb {Z }}^{r \time t}\)在对角线上。在本文中，我们改进了 Hemmecke 和 Schultz 从 2003 年开始的算法结果 [Hemmecke 和 Schultz，Math. 编。2003]解决两阶段随机IP。该算法基于 Graver 增强框架，我们的主要贡献是对增强步骤的大小给出明确的双指数界限。增广步骤大小的先前界限依赖于来自交换代数的非建设性有限性参数，因此只知道一个隐式界限，它取决于参数r、  s、  t和\(\Delta \)，其中\(\Delta \)是约束矩阵的最大条目。然而，我们新的改进界限是通过一个关于向量空间中路径交叉点的新定理获得的。由于我们的新界限，我们获得了一个算法来解决时间\(f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)\) 的2 阶段随机 IP ，其中f是双指数函数。为了补充我们的结果，我们还证明了增广步骤大小的双指数下限。