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 \cT x A x = b, ≤ x ≤ u, x ∈ Zr + ns \ where the constraint matrix A ∈ Znt × r +ns consists of n matrices Ai ∈ Zt × r on the vertical line and n matrices Bi ∈ Zt × 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 ≤ γ satisfying z2 α β for given α, β, γ ∈ 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 β 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 22δ(s+t) |I|O(1) for some δ > 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||∞, ||c||∞, ||||∞ and the largest absolute value in the constraint matrix 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 A.