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 $\min \{c^T x \mid \mathcal {A} x = b, \ell \le x \le u, x \in \mathbb {Z}^{r + ns} \}$ 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 $z \le \gamma$ 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 by proving that the problem remains NP-hard, even if each primenumber 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 $\varDelta$ 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 $\mathcal A$.

This work was supported by DFG project JA 612/20-1.