Consider a high-multiplicity Bin Packing instance $I$ with $d$ distinct item types. In 2014, Goemans and Rothvoss gave an algorithm with runtime ${{|I|}^2}^{O(d)}$ for this problem~[SODA'14], where $|I|$ denotes the encoding length of the instance $I$. Although Jansen and Klein~[SODA'17] later developed an algorithm that improves upon this runtime in a special case, it has remained a major open problem by Goemans and Rothvoss~[J.ACM'20] whether the doubly exponential dependency on $d$ is necessary. We solve this open problem by showing that unless the ETH fails, there is no algorithm solving the high-multiplicity Bin Packing problem in time ${{|I|}^2}^{o(d)}$. To prove this, we introduce a novel reduction from 3-SAT. The core of our construction is efficiently encoding all information from a 3-SAT instance with $n$ variables into an ILP with $O(\log(n))$ variables and constraints. This result confirms that the Goemans and Rothvoss algorithm is essentially best-possible for Bin Packing parameterized by the number $d$ of item sizes in the context of XP time algorithms.

In the problem of Parallel Task Scheduling (PTS), we are asked to schedule n jobs, each with a fixed processing time and machine requirement, such that the completion time of the last job is minimized. Jansen and Rau (2019) presented an algorithm for PTS that achieves an approximation ratio of (3/2)OPT + pmax. They additionally posed the open question whether an approximation ratio of (4/3)OPT + pmax is possible. In this work, we present such an algorithm with a running time of O(n log n).

Block-structured integer linear programs (ILPs) play an important role in various application fields. We address n-fold ILPs where the matrix 𝒜 has a specific structure, i.e., where the blocks in the lower part of 𝒜 consist only of the row vectors (1,… ,1). In this paper, we propose an approach tailored to exactly these combinatorial n-folds. We utilize a divide and conquer approach to separate the original problem such that the right-hand side iteratively decreases in size. We show that this decrease in size can be calculated such that we only need to consider a bounded amount of possible right-hand sides. This, in turn, lets us efficiently combine solutions of the smaller right-hand sides to solve the original problem. We can decide the feasibility of, and also optimally solve, such problems in time (n r Δ)^O(r) log(‖b‖_∞), where n is the number of blocks, r the number of rows in the upper blocks and Δ = ‖A‖_∞. We complement the algorithm by discussing applications of the n-fold ILPs with the specific structure we require. We consider the problems of (i) scheduling on uniform machines, (ii) closest string and (iii) (graph) imbalance. Regarding (i), our algorithm results in running times of p_max^O(d)|I|^O(1), matching a lower bound derived via ETH. For (ii) we achieve running times matching the current state-of-the-art in the general case. In contrast to the state-of-the-art, our result can leverage a bounded number of column-types to yield an improved running time. For (iii), we improve the parameter dependency on the size of the vertex cover.