The Double Exponential Runtime is Tight for 2-Stage Stochastic ILPs
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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 |𝒜 x = b, ℓ≤ x ≤ u, x ∈ℤ^r + ns} where the constraint matrix 𝒜∈ℤ^nt × r +ns consists of n matrices A_i ∈ℤ^t × r on the vertical line and n matrices B_i ∈ℤ^t × 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 z^2 ≡αβ 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 2^2^δ(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||_∞, ||ℓ||_∞ and the largest absolute value Δ 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.
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