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. 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 proving that the problem remains NP-hard, even if each prime number only occurs constantly often. Using this new hardness result for the textsc {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 constraint matrix is the transpose of {mathcal {A}}.