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

Abstract: 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 $z² \equiv \alpha \bmod \beta$ 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 $$2^{{2^{{\delta(s+t)}}}} |I|<sup>{O(1)}$$ for some $\delta &gt; 0$ 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 $r$, $||b||<em>{\infty}$, $||c||</em>{\infty}, ||\ell||_{\infty}$ 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 $\mathcal A$.