两阶段随机ILP的双指数紧性
The Double Exponential Runtime is Tight for 2-Stage Stochastic ILPs
摘要：
我们考虑基本的算法数论问题和它们的关系，一类块结构的线性规划（ILP）称为$2$阶段随机。$2$阶段随机ILP是$\min \{c^T x \mid \mathcal{A} x = b, \ell \leq x \leq u, x \in \mathbb{Z}^{r + ns} \}$形式的整数规划，其中约束矩阵$\mathcal{A} \in \mathbb{Z}^{nt \times r +ns}$由垂直线上的$n$矩阵$A_i \in \mathbb{Z}^{t \times r}$和对角线上的$n$矩阵$B_i \in \mathbb{Z}^{t \times s}$组成.首先，我们展示了一个更强的硬度结果称为二次同余的数论问题，其目标是计算一个数$z \leq γ$满足$z^2 \equiv α\bmod β$给定的$α, β, γ\in \mathbb{Z}$。Manders和Adleman在1978年已经证明了这个问题是NP难的。然而，这种困难仅适用于$β$的素数分解允许每个素数的大重数的情况。我们绕过了这个必要性，证明问题仍然是NP难的，即使每个素数只是经常出现。然后，使用这个新的硬度结果的二次同余问题，我们证明了一个下界的$2^{2^{δ(s+t)}} |I|^{O(1)}$的一些$δ> 0$的运行时间的任何算法解决$2$阶段的随机ILP假设的指数时间假设（ETH）。这里，$|I|$是实例的编码长度。如果约束矩阵$\mathcal{A}$中的$r$、$||b||_{\infty}$、$||c||_{\infty}, ||\ell||_{\infty}$和最大绝对值$Δ$恒定，则该结果也成立。这表明最先进的算法几乎是紧密的。此外，它证明了这样的怀疑，这些ILP确实比密切相关的$n$ -倍ILP更难解决，其中约束矩阵是$\mathcal A$的转置。
英文摘要：
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 $z \leq γ$ satisfying $z^2 \equiv α\bmod β$ for given $α, β, γ\in \mathbb{Z}$. 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 $β$ 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 $2^{2^{δ(s+t)}} |I|^{O(1)}$ for some $δ> 0$ 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 $Δ$ 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 $\mathcal A$.
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