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.