We consider so-called $N$-fold integer programs (IPs) of the form $\max\{c^T x : Ax = b, \ell \leq x \leq u, x \in \mathbb Z^{nt}\}, where $A \in \mathbb Z^{(r+sn)\times nt} consists of $n$ arbitrary matrices $A^{(i)} \in \mathbb Z^{r\times t}$ on a horizontal, and $n$ arbitrary matrices $B^{(j)} \in \mathbb Z^{s\times t} on a diagonal line. Several recent works design fixed-parameter algorithms for $N$-fold IPs by taking as parameters the numbers of rows and columns of the $A$- and $B$-matrices, together with the largest absolute value $\Delta$ over their entries. These advances provide fast algorithms for several well-studied combinatorial optimization problems on strings, on graphs, and in machine scheduling. In this work, we extend this research by proposing algorithms that additionally harness a partition structure of submatrices $A^{(i)}$ and $B^{(j)}$, where row indices of non-zero entries do not overlap between any two sets in the partition. Our main result is an algorithm for solving any $N$-fold IP in time $nt log(nt)L^2(S_A)^{O(r+s)}(p_Ap_B\Delta)^{O(rp_Ap_B+sp_Ap_B)}$, where $p_A$ and $p_B$ are the size of the largest set in such a partition of $A^{(i)}$ and $B^{(j)}$, respectively, $S_A$ is the number of parts in the partition of $A = (A^{(1)},..., A^{(n)}), and $L = (log(||u - \ell||_\infty)\cdot (log(max_{x:\ell \leq x \leq u} |c^Tx|))$ is a measure of the input. We show that these new structural parameters are naturally small in high-multiplicity scheduling problems, such as makespan minimization on related and unrelated machines, with and without release times, the Santa Claus objective, and the weighted sum of completion times. In essence, we obtain algorithms that are exponentially faster than previous works by Knop et al. (ESA 2017) and Eisenbrand et al./Kouteck{\'y} et al. (ICALP 2018) in terms of the number of job types.

翻译：我们考虑形如 $\max\{c^T x : Ax = b, \ell \leq x \leq u, x \in \mathbb Z^{nt}\}$ 的所谓 $N$-折叠整数规划（IP），其中 $A \in \mathbb Z^{(r+sn)\times nt}$ 由水平方向上的 $n$ 个任意矩阵 $A^{(i)} \in \mathbb Z^{r\times t}$ 和对角线上的 $n$ 个任意矩阵 $B^{(j)} \in \mathbb Z^{s\times t}$ 构成。最近的几项工作通过将 $A$ 矩阵和 $B$ 矩阵的行数与列数以及其元素的最大绝对值 $\Delta$ 作为参数，设计了 $N$-折叠 IP 的固定参数算法。这些进展为字符串、图以及机器调度中几个经过充分研究的组合优化问题提供了快速算法。在本工作中，我们通过提出额外利用子矩阵 $A^{(i)}$ 和 $B^{(j)}$ 划分结构的算法来扩展这一研究，其中非零元素的行索引在划分的任何两个集合之间不重叠。我们的主要结果是求解任意 $N$-折叠 IP 的算法，其时间复杂度为 $nt log(nt)L^2(S_A)^{O(r+s)}(p_Ap_B\Delta)^{O(rp_Ap_B+sp_Ap_B)}$，其中 $p_A$ 和 $p_B$ 分别是 $A^{(i)}$ 和 $B^{(j)}$ 此类划分中最大集合的大小，$S_A$ 是 $A = (A^{(1)},..., A^{(n)})$ 划分的部件数，$L = (log(||u - \ell||_\infty)\cdot (log(max_{x:\ell \leq x \leq u} |c^Tx|))$ 是输入的一种度量。我们证明，这些新的结构参数在高重数调度问题中自然很小，例如在相关和不相关机器上的完工时间最小化问题（含或不含释放时间）、Santa Claus 目标函数以及加权完成时间和问题。本质上，我们获得的算法在作业类型数量方面，比 Knop 等人（ESA 2017）以及 Eisenbrand 等人/Kouteck{\'y} 等人（ICALP 2018）先前的工作指数级更快。