We study an important case of ILPs max{c T x | Ax = b, l ≤ x ≤ u, x ∈ Z nt } with n • t variables and lower and upper bounds , u ∈ Z nt. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s × t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and ∆, where ∆ is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Ω(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs∆) O(r 2 s+s 2) L 2 • nt log O(1) (nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L.

We study an important case of ILPs max{c T x | Ax = b, l ≤ x ≤ u, x ∈ Z nt } with n • t variables and lower and upper bounds ℓ, u ∈ Z nt . In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s × t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and ∆, where ∆ is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Ω(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs∆) O(r 2 s+s 2 ) L 2 • nt log O(1) (nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L. ACM Subject Classification Mathematics of computing → Integer programming Keywords and phrasesNear-Linear Time Algorithm, n-fold, Color Coding 1 * This work was partially supported by DFG Project "Strukturaussagen und deren Anwendung in Scheduling-und Packungsprobleme", JA 612/20-1 XX:2 Near-Linear Time Algorithm for n-fold ILPs via Color Coding Summary of ResultsWe present an algorithm, which solves n-fold ILPs in time (rs∆) O(r 2 s+s 2 ) L • nt log 4 (nt) + LP, where LP is the time to solve the LP relaxation of the n-fold. This is the first algorithm with a near-linear dependence on the number of variables. The crucial step is to speed up the computation of the improving directions. We circumvent the need for solving the LP relaxation. This leads to a purely combinatorial algorithm with running time (rs∆) O(r 2 s+s 2 ) L 2 • nt log 6 (nt).

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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 | Ax = b, ≤ 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. 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 ≡ α mod 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. 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|| ∞ , || || ∞ 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

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