In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) - so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form min {c^⊺x∣Ax = b, x ≥ 0, x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each stage there are larger blocks with few columns connecting the blocks in a tree like fashion. Over the last years there was enormous progress in the area of block structured IPs. For many of the known block IP classes - such as n-fold, tree-fold, and two-stage stochastic IPs, nearly matching upper and lower bounds are known concerning their computational complexity. One of the major gaps that remained however was the parameter dependency in the running time for an algorithm solving multistage stochastic IPs. Previous algorithms require a tower of t exponentials, where t is the number of stages. In contrast, only a double exponential lower bound was known based on the exponential time hypothesis. In this paper we show that the tower of t exponentials is actually not necessary. We show an improved running time of \(2^{(d\left\Vert A \right\Vert _\infty)^{\mathcal {O}(d^{3t+1})}} \cdot rn\log ^{\mathcal {O}(2^d)}(rn) \) for the algorithm solving multistage stochastic IPs, where d is the sum of columns in the connecting blocks and rn is the number of rows. Hence, we obtain the first bound by an elementary function for the running time of an algorithm solving multistage stochastic IPs. In contrast to previous works, our algorithm has only a triple exponential dependency on the parameters and only doubly exponential for every constant t. By this we come very close to the known double exponential bound that holds already for two-stage stochastic IPs, i.e. multistage stochastic IPs with two stages.

The improved running time of the algorithm is based on new bounds for the proximity of multistage stochastic IPs. The idea behind the bound is based on generalization of a structural lemma originally used for two-stage stochastic IPs. While the structural lemma requires iteration to be applied to multistage stochastic IPs, our generalization directly applies to inherent combinatorial properties of multiple stages. Already a special case of our lemma yields an improved bound for the Graver complexity of multistage stochastic IPs.

在本文中，我们研究了求解一类块结构整数程序 (IP)（即所谓的多级随机 IP）的计算复杂性。多级随机 IP 是 min { c ^⊺ x ∣ Ax = b ,  x ≥ 0 ,  x积分}形式的 IP  ，其中约束矩阵A由在对角线上排序的小块矩阵组成，并且每个阶段都有较大的块很少有柱子像时尚一样将各个块连接成树状。在过去的几年里，块结构 IP 领域取得了巨大的进步。对于许多已知的块 IP 类别 - 例如n折叠、树折叠和两阶段随机 IP，其计算复杂度几乎匹配上限和下限。然而，仍然存在的主要差距之一是求解多级随机 IP 的算法运行时间的参数依赖性。以前的算法需要一个由t个指数组成的塔，其中t是级数。相反，根据指数时间假设，仅知道双指数下限。在本文中，我们证明了t指数塔实际上是不必要的。我们展示了 \(2^{(d\left\Vert A \right\Vert _\infty)^{\mathcal {O}(d^{3t+1})}} \cdot rn\log 的改进运行时间^{\mathcal {O}(2^d)}(rn) \) 用于求解多级随机 IP 的算法，其中d是连接块中的列数之和，rn是行数。因此，我们获得了求解多级随机 IP 的算法运行时间的初等函数的第一个界限。与以前的工作相比，我们的算法对参数仅具有三重指数依赖性，并且对于每个常数t仅具有双重指数依赖性。由此，我们非常接近已知的双指数界限，该界限已经适用于两阶段随机 IP，即具有两阶段的多阶段随机 IP。

该算法运行时间的改进基于多级随机 IP 邻近度的新界限。界限背后的想法基于最初用于两阶段随机 IP 的结构引理的概括。虽然结构引理要求将迭代应用于多级随机 IP，但我们的概括直接应用于多级的固有组合属性。我们的引理的一个特例已经为多级随机 IP 的 Graver 复杂性提供了改进的界限。