We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called $2$-stage stochastic. A $2$-stage stochastic ILP is an integer program of the form $\min \{c^T x \mid \mathcal{A} x = b, \ell \leq x \leq u, x \in \mathbb{Z}^{r + ns} \}$ where the constraint matrix $\mathcal{A} \in \mathbb{Z}^{nt \times r +ns}$ consists of $n$ matrices $A_i \in \mathbb{Z}^{t \times r}$ on the vertical line and $n$ matrices $B_i \in \mathbb{Z}^{t \times s}$ on the diagonal line aside. First, we show a stronger hardness result for a number theoretic problem called Quadratic Congruences where the objective is to compute a number $z \leq \gamma$ satisfying $z^2 \equiv \alpha \bmod \beta$ for given $\alpha, \beta, \gamma \in \mathbb{Z}$. This problem was proven to be NP-hard already in 1978 by Manders and Adleman. However, this hardness only applies for instances where the prime factorization of $\beta$ admits large multiplicities of each prime number. We circumvent this necessity proving that the problem remains NP-hard, even if each prime number only occurs constantly often. Then, using this new hardness result for the Quadratic Congruences problem, we prove a lower bound of $2^{2^{\delta(s+t)}} |I|^{O(1)}$ for some $\delta > 0$ for the running time of any algorithm solving $2$-stage stochastic ILPs assuming the Exponential Time Hypothesis (ETH). Here, $|I|$ is the encoding length of the instance. This result even holds if $r$, $||b||_{\infty}$, $||c||_{\infty}, ||\ell||_{\infty}$ and the largest absolute value $\Delta$ in the constraint matrix $\mathcal{A}$ are constant. This shows that the state-of-the-art algorithms are nearly tight. Further, it proves the suspicion that these ILPs are indeed harder to solve than the closely related $n$-fold ILPs where the contraint matrix is the transpose of $\mathcal A$.

中文翻译：

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两阶段随机 ILP 的双指数运行时间很紧

我们考虑基本的算法数论问题及其与一类称为 $2$-stage stochastic 的块结构整数线性规划 (ILP) 的关系。$2$-stage 随机 ILP 是形式为 $\min \{c^T x \mid \mathcal{A} x = b, \ell \leq x \leq u, x \in \mathbb{Z }^{r + ns} \}$ 其中约束矩阵 $\mathcal{A} \in \mathbb{Z}^{nt \times r +ns}$ 由 $n$ 个矩阵 $A_i \in \mathbb{ Z}^{t \times r}$ 在垂直线上，$n$ 矩阵 $B_i \in \mathbb{Z}^{t \times s}$ 在对角线上。首先，我们为称为二次同余的数论问题展示了更强的硬度结果，其中目标是计算一个数 $z \leq \gamma$ 满足 $z^2 \equiv \alpha \bmod \beta$ 对于给定的 $\alpha , \beta, \gamma \in \mathbb{Z}$。这个问题在 1978 年就被 Manders 和 Adleman 证明是 NP-hard 问题。然而，这种难度仅适用于 $\beta$ 的素数分解允许每个素数有很大的多重性的情况。我们绕过了这种必要性，证明问题仍然是 NP-hard 问题，即使每个素数只是经常出现。然后，使用二次同余问题的新硬度结果，我们证明了 $2^{2^{\delta(s+t)}} |I|^{O(1)}$ 的下界 $\delta > 0$ 用于解决 $2$-stage 随机 ILPs 假设指数时间假设 (ETH) 的任何算法的运行时间。其中，$|I|$ 是实例的编码长度。这个结果甚至成立如果 $r$, $||b||_{\infty}$, $||c||_{\infty}, ||\ell||_{\infty}$ 和最大绝对值约束矩阵 $\mathcal{A}$ 中的值 $\Delta$ 是常数。这表明最先进的算法几乎是紧凑的。此外，它证明了这些 ILP 确实比密切相关的 $n$-fold ILP 更难解决的怀疑，其中约束矩阵是 $\mathcal A$ 的转置。