Block-structured integer linear programs (ILPs) play an important role in various application fields. We address $n$-fold ILPs where the matrix $\mathcal{A}$ has a specific structure, i.e., where the blocks in the lower part of $\mathcal{A}$ consist only of the row vectors $(1,\dots,1)$. 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 $(n r \Delta)^{O(r)} \log(\|b\|_\infty),$ where $n$ is the number of blocks, $r$ the number of rows in the upper blocks and $\Delta=\|A\|_\infty$. 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.

翻译：块结构整数线性规划（ILP）在多个应用领域中具有重要作用。本文研究$n$-折叠ILP问题，其中矩阵$\mathcal{A}$具有特定结构，即$\mathcal{A}$下部的块仅由行向量$(1,\dots,1)$构成。我们提出一种专门针对此类组合$n$-折叠问题的求解方法。采用分治策略对原问题进行分解，使右侧向量在迭代过程中规模递减。我们证明这种规模缩减可通过计算实现，从而仅需考虑有限数量的可能右侧向量。这使得我们能够高效整合较小右侧向量子问题的解以求解原问题。我们可在$(n r \Delta)^{O(r)} \log(\|b\|_\infty)$时间内判定此类问题的可行性并求得最优解，其中$n$为块数，$r$为上部块的行数，$\Delta=\|A\|_\infty$。我们进一步讨论了具有特定结构的$n$-折叠ILP的应用场景，包括：（i）均匀机器调度问题，（ii）最近字符串问题，以及（iii）（图）不平衡问题。对于问题（i），算法获得$p_{\max}^{O(d)}|I|^{O(1)}$的时间复杂度，与基于指数时间假说（ETH）推导的下界相匹配。对于问题（ii），在一般情形下达到与当前最优算法相当的时间复杂度。与现有技术相比，我们的方法能够利用有限列类型的特性获得更优的时间复杂度。对于问题（iii），我们改进了关于顶点覆盖规模的参数依赖关系。