We study two classic variants of block-structured integer programming. Two-stage stochastic programs are integer programs of the form $\{A_i \mathbf{x} + D_i \mathbf{y}_i = \mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where $A_i$ and $D_i$ are bounded-size matrices. On the other hand, $n$-fold programs are integer programs of the form $\{{\sum_{i=1}^n C_i\mathbf{y}_i=\mathbf{a}} \textrm{ and } D_i\mathbf{y}_i=\mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where again $C_i$ and $D_i$ are bounded-size matrices. It is known that solving these kind of programs is fixed-parameter tractable when parameterized by the maximum dimension among the relevant matrices $A_i,C_i,D_i$ and the maximum absolute value of any entry appearing in the constraint matrix. We show that the parameterized tractability results for two-stage stochastic and $n$-fold programs persist even when one allows large entries in the global part of the program. More precisely, we prove that: - The feasibility problem for two-stage stochastic programs is fixed-parameter tractable when parameterized by the dimensions of matrices $A_i,D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we allow matrices $A_i$ to have arbitrarily large entries. - The linear optimization problem for $n$-fold integer programs that are uniform -- all matrices $C_i$ are equal -- is fixed-parameter tractable when parameterized by the dimensions of matrices $C_i$ and $D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we require that $C_i=C$ for all $i=1,\ldots,n$, but we allow $C$ to have arbitrarily large entries. In the second result, the uniformity assumption is necessary; otherwise the problem is $\mathsf{NP}$-hard already when the parameters take constant values. Both our algorithms are weakly polynomial: the running time is measured in the total bitsize of the input.

翻译：我们研究了块结构整数规划的两种经典变体。两阶段随机规划是形如 $\{A_i \mathbf{x} + D_i \mathbf{y}_i = \mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$ 的整数规划，其中 $A_i$ 和 $D_i$ 是有界尺寸的矩阵。另一方面，$n$ 折规划是形如 $\{{\sum_{i=1}^n C_i\mathbf{y}_i=\mathbf{a}} \textrm{ and } D_i\mathbf{y}_i=\mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$ 的整数规划，其中 $C_i$ 和 $D_i$ 同样是有界尺寸的矩阵。已知当以相关矩阵 $A_i,C_i,D_i$ 的最大维度以及约束矩阵中任何条目的最大绝对值为参数时，求解这类问题是固定参数可处理的。我们证明，即使允许程序的全局部分出现大条目，两阶段随机规划和 $n$ 折规划的参数可处理性结果仍然成立。更精确地说，我们证明：- 两阶段随机规划的可行性问题在以矩阵 $A_i,D_i$ 的维度和矩阵 $D_i$ 条目的最大绝对值为参数时是固定参数可处理的。即，我们允许矩阵 $A_i$ 具有任意大的条目。- 均匀 $n$ 折整数规划的线性优化问题（所有矩阵 $C_i$ 相等）在以矩阵 $C_i$ 和 $D_i$ 的维度以及矩阵 $D_i$ 条目的最大绝对值为参数时是固定参数可处理的。即，我们要求对所有 $i=1,\ldots,n$ 有 $C_i=C$，但允许 $C$ 具有任意大的条目。在第二个结果中，均匀性假设是必要的；否则，当参数取常数值时，问题已经是 $\mathsf{NP}$-难问题。我们的两个算法都是弱多项式算法：运行时间按输入的总比特大小度量。