The Double Exponential Runtime is Tight for 2-Stage Stochastic ILPs

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 $Ai \in \mathbb{Z}^{t \times r}$ on the vertical line and $n$ matrices $Bi \in \mathbb{Z}^{t \times s}$ 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 $z² \equiv \alpha \bmod \beta$ 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 $\mathcal A$.